Retrieving information about a group from its character table

Retrieving information about a group from its character table Sandro Mattarei

Thesis submitted for the degree of Doctor of Philosophy of the University of Warwick Mathematics Institute University of Warwick Coventry CV4 7AL July 1992

Contents 1 Introduction

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2 Technical results 2.1 Commutators . . . . . . . . . . . . . . 2.2 Linear maps and commutation . . . . . 2.3 Irreducible modules for abelian groups 2.4 Homogeneous modules . . . . . . . . . 2.5 Induction and tensor induction . . . . 2.6 Basic commutators . . . . . . . . . . . 2.7 Character table isomorphisms . . . . .

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12 12 13 17 23 25 30 35

3 Looking for a counterexample 3.1 Introduction . . . . . . . . . . . . . . . 3.2 Structure of a minimal counterexample 3.3 Chief factors . . . . . . . . . . . . . . . ¯ on H 0 . . . 3.4 More about the action of Q 3.5 The smallest cases which can occur . .

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37 37 38 40 46 50

4 Comparing character tables 4.1 Our philosophy . . . . . . . . . . . . 4.2 Vanishing of character values . . . . 4.3 A tool for comparing character tables 4.4 A generalization . . . . . . . . . . . .

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56 56 61 63 68

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5 Counterexamples 72 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 General construction . . . . . . . . . . . . . . . . . . . . . . . 74 1

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CONTENTS 5.3 5.4 5.5 5.6 5.7

The unary case . . . . . . . . . . . . . . . Matrix representations for the unary case . The binary case . . . . . . . . . . . . . . . Matrix representations for the binary case Power-maps . . . . . . . . . . . . . . . . .

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6 Nilpotent counterexamples 7 Wreath products 7.1 Characters of wreath products . . . . 7.2 Conjugacy classes of wreath products 7.3 Character tables of wreath products . 7.4 An application . . . . . . . . . . . .

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80 86 92 99 103 109

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119 . 119 . 127 . 130 . 136

Acknowledgements I would like to thank my research supervisor Dr T. O. Hawkes for his constant support. I acknowledge the financial support of CNR (Consiglio Nazionale delle Ricerche, Italy).

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Declaration The work contained in this thesis is original except where otherwise indicated.

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Summary This thesis concerns character tables of finite soluble groups. In particular, our main objective is that of showing that the derived length of a soluble group G is not determined by the character table of G. In fact, in Chapters 5 and 6 we shall construct pairs (G, H) of groups which have identical character tables but different derived lengths, namely 2 and 3. A more general result will be proved in Chapter 7, namely that for any natural number n ≥ 2, there exist pairs (G, H) of groups with identical character tables, and derived lengths n and n + 1 respectively. Two by-products of our investigation are a method for comparing character tables in special situations (in Chapter 4), and a description of the character tables of wreath products (in Chapter 7).

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Chapter 1 Introduction Representation theory, and in particular character theory, have proved to be powerful tools for the study of finite groups. Furthermore, character theory provides a practical way of gathering a lot of information about a group G in a very condensed form, by means of a matrix with complex entries, called the character table of G. This is especially true for simple groups. In fact, character tables are perhaps the main information provided by the Atlas of finite simple groups [4], which is an indispensable reference for the classification of the finite simple groups. Each finite simple group is uniquely identified by its character table, and some sporadic simple groups were known through their character tables even before their existence was proved. The character table of a finite group G is the matrix T (which turns out to be square), whose (i, j)th entry is χi (gj ), where χ1 , . . . , χk are the irreducible characters of G (over the complex field), and g1 , . . . , gk are a set of representatives for the conjugacy classes K1 , . . . , Kk of G (with gj ∈ Kj ). Since characters are class functions, the character table T is not affected by the choice of different representatives gj0 ∈ Kj , and thus the columns of T will also be indexed by the conjugacy classes of G. It also follows from this that the knowledge of the character table T of G amounts to the knowledge of all irreducible characters of G, as functions from G into C, once one knows the correspondence between rows of T and irreducible characters of G, and the correspondence between columns of T and conjugacy classes of G. These correspondences will not be considered part of the object character table, nor will any other information about G and its characters, like orders of the elements, or Frobenius-Schur indicators. An additional piece of information, 6

7 namely the so-called power-maps, will be considered occasionally, but this will be explicitly stated as character table with powermaps. Unlike simple groups, soluble groups are not uniquely identified by their character tables. The easiest example is given by the two non-abelian groups of order eight, namely the dihedral group D8 and the quaternion group Q8 ; in fact, their common character table is the following matrix T .     T =   



1 1 1 1 1  1 1 1 −1 −1   1 1 −1 1 −1   1 1 −1 −1 1   2 −2 0 0 0

We may notice that the first row of T corresponds to the trivial character of G, and the first column corresponds to the identity class of G. Apart from this, which we shall adopt as a convention, there is no natural rule for ordering the conjugacy classes of G and its irreducible characters (though practical rules are used in [4], for the sake of convenience). Consequently, the character table T of a group G is defined up to permutations of its rows and of its columns. Hence, we shall say that the character tables T1 and T2 of two groups are identical if it is possible to obtain T2 from T1 by permuting rows and columns of T1 . Later we shall give a more handy definition of having identical character tables, namely Definition 2.7.1. We shall see in Chapter 4 that D8 and Q8 form just a special case of a more general situation in which two groups have identical character tables. Although a group G is not uniquely determined by its character table T , a lot of properties of G can be read off from T . We shall give a brief review of some of these properties, after noticing that each property can usually be obtained from T in several ways. A first class of properties employs the so-called second orthogonality relation, part of which is the formula |CG (gj )| =

k X

|χi (gj )|2 ,

i=1

which allows one to compute the order of the centralizer of an element of the class Kj by means of the corresponding column of T . In particular, for the

8

CHAPTER 1. INTRODUCTION

identity class K1 , the above formula becomes |G| =

k X

χi (1)2 ,

i=1

and thus yields the order of G. As a consequence, the length of each conjugacy class Kj of G can be computed, because |Kj | = |G : CG (gj )|. Also, the columns of T which correspond to central elements of G can be determined, and thus the order of the centre of G can be computed. It follows that it can be decided whether G is abelian (though a simpler method for this is checking that all characters of G have degree 1). If this is the case, then G is determined by T up to isomorphism. More generally, Z(G) is always determined by T up to isomorphism. The second class of properties which we are going to examine employs the fact that the kernels of the irreducible characters can be read off from the character table (meaning that it can be decided which classes of G are contained in the kernel of a given irreducible character); in fact, it is easy to see that ker χ = {g ∈ G | χ(g) = χ(1)}, if χ ∈ Irr(G). Now, kernels of characters are normal subgroups of G, and conversely, each normal subgroup N of G is the kernel of some character of G, for instance the character afforded by the regular representation of G/N , viewed as a representation of G. Since the kernel of a reducible character is the intersection of the kernels of its irreducible constituents, it follows that all normal subgroups of G can be found from T , as intersections of kernels of irreducible characters. Moreover, since each normal subgroup N of G is a union of conjugacy classes of G, its order can be computed, and inclusion relations with other normal subgroups can be determined. To summarize, from T we read the lattice of normal subgroups of G, each with its order attached. But we can do more: for each normal subgroup N , the character table of the factor group G/N can be extracted from T , simply by deleting those rows of T which correspond to irreducible characters χ such that N 6≤ ker χ, and then replacing each set of identical columns with a single one. Let us remark that a similar procedure for obtaining the character table of the normal subgroup N does not exist. For instance, D8 has exactly three normal subgroups of order 4, which cannot be distinguished by looking at

9 the character table of D8 ; yet they do not have identical character tables, because one of them is cyclic, and the other two are elementary abelian. The knowledge of the lattice of normal subgroups of G, together with the character tables of the corresponding factor groups, allows one to decide about the nilpotency, supersolubility, or solubility of G. In fact, G is nilpotent, respectively supersoluble, or soluble, if and only if there is a normal series 1 = N0 < N1 < · · · < Ns = G, such that all factors Ni /Ni−1 are central, respectively cyclic of prime order, or p-groups; all of these conditions can be checked on the character table T of G. The terms of the upper central series of G can be inductively located in T , by taking centres and factor groups in turns; in particular, if G is nilpotent, its nilpotency class is determined by T . The lower central series of G can also be found, for instance by inspecting all central series descending from G and finding the fastest descending one. However, another method is available, which yields even more. In fact, it follows by induction from [13, Problem 3.10(a)] that the character table of G allows one to decide which conjugacy classes of G contain elements of the form g = [x1 , . . . , xi ] with x1 , . . . , xi ∈ G; these conjugacy classes generate γi (G), the ith term of the lower central series, though their (set-theoretical) union may be properly contained in γi (G). Inspection of the lattice of normal subgroups of G (with orders) shows which normal subgroups N are nilpotent: they are exactly those which contain normal subgroups of G of order |N |p for all prime divisors p of |N | (here, as usual, |N |p denotes the biggest power of p which divides the order of N ). As a consequence, the Fitting subgroup of G can be found (as the biggest nilpotent normal subgroup of G); hence, if G is soluble, the Fitting series of G can be determined inductively, and the Fitting length can be computed. What about the derived series of G (and in particular the derived length of G, if G is soluble)? The derived subgroup G0 can certainly be read off from T , as the smallest normal subgroup N of G such that G/N is abelian, or equivalently as the intersection of the kernels of all linear characters of G. The problem of finding the second derived subgroup G00 amounts to being able to tell whether G0 is abelian from the character table of G. The following more general question appeared as Problem 10 in R. Brauer’s report

10

CHAPTER 1. INTRODUCTION

on representations of finite groups, in [19, page 141]: Given the character table of a group G and the set of conjugacy classes of G which make up a normal subgroup N of G, can it be decided whether or not N is abelian? As Brauer then remarked, a positive answer to this question would allow one to identify the terms of the derived series of G by looking at its character table, and in particular to compute the derived length of G, for G soluble. Unfortunately, the answer to Brauer’s Problem 10 is negative, as announced by A. I. Saksonov in [20]. A computational approach to this problem has been taken recently by K. Dockx and P. Igodt in [7], which led to the same conclusion and produced additional examples. However, neither Saksonov, nor Dockx and Igodt, answered Brauer’s question about the derived length. One of the main results in this thesis is the construction of groups G and H with identical character tables and derived lengths 2 and 3 respectively, which proves that the derived length of a soluble group cannot be read off from its character table. This will be done in Chapter 5. The discovery of the above mentioned examples was a consequence of the close study of the structure of a minimal example of groups with identical character tables and different derived lengths. This study is also part of this thesis, and will be carried out in Chapter 3. Chapter 6 is devoted to the construction of another example of groups with identical character tables and derived lengths 2 and 3. The groups of this example are p-groups, unlike those of Chapter 5, which are not nilpotent. The existence of this chapter is justified by the fact that the discussion of a minimal example in Chapter 3 is carried out under the assumption that the groups in question are not nilpotent. A tool for the comparison of character tables will be developed in Chapter 4. Being suited to our examples of Chapters 5 and 6, it concerns a rather special configuration, that of Camina groups. However, since Camina groups have been studied extensively, and seem to arise in many different situations, the results of this chapter may prove useful elsewhere. Early and shorter versions of Chapters 4 and 5 will appear together as an article (namely [17]), in the Journal of the London Mathematical Society. The final chapter of this thesis, namely Chapter 7, concerns character tables of wreath products. We shall prove that the character table of a

11 wreath product G o A is completely determined by the permutation group A and the character table of G. An almost immediate consequence of this fact is the construction of pairs (G, H) of groups with identical character tables and derived lengths n and n + 1, for any given integer n ≥ 2. Some more or less standard results from group theory and representation theory, which we shall need, are collected in Chapter 2.

Chapter 2 Technical results 2.1

Commutators

We shall use the standard notation of [10] for commutators. In particular, if A and B are subsets of a group G, we set [A, B] = h[a, b] | a ∈ A, b ∈ Bi. However, it will be useful to have a notation also for the set of commutators [a, b] with a ∈ A and b ∈ B. Thus, we shall occasionally use the following non-standard notation: bA, Bc = {[a, b] | a ∈ A, b ∈ B}. If G1 , . . . , Gn are subsets of G, we set [G1 , . . . , Gn ] = h[g1 , . . . , gn ] | gi ∈ Gi i, where [g1 , . . . , gn ] is defined recursively by the formula [g1 , . . . , gn ] = [[g1 , . . . , gn−1 ], gn ]. We observe that [G1 , . . . , Gn ] ≤ [. . . [[G1 , G2 , ], G3 ], . . . , Gn ], though equality does not hold in general. We shall need the following well-known lemma about coprime actions. 12

2.2. LINEAR MAPS AND COMMUTATION

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Lemma 2.1.1 Let A be a Q-group, that is to say, let Q be a group of operators for the group A, and suppose that (|A|, |Q|) = 1. Then [[A, Q], Q] = [A, Q], and A = [A, Q]CA (Q). Furthermore, if A is abelian, then A = [A, Q] × CA (Q). Proof Our first statement is [10, Kapitel III, Hilfssatz 13.3 b)], from whose proof our second statement follows. Our third statement is [10, Kapitel III, Satz 13.4 b)]. ¤

2.2

Linear and bilinear maps arising from commutation

Lemma 2.2.1 Suppose that H1 , H2 , H3 , K1 , K2 , K3 are subgroups of a group G. Suppose that Ki C Hi and that Hi /Ki is abelian (i = 1, 2, 3). Suppose also that [H1 , H2 ] ≤ H3 and that [H1 , K2 ], [K1 , H2 ], [H1 , H2 , H1 ] and [H1 , H2 , H2 ] are all contained in K3 . Then there exists a Z-bilinear map γ : H1 /K1 × H2 /K2 → H3 /K3 such that (xK1 , yK2 )γ = [x, y]K3 for all x ∈ H1 and for all y ∈ H2 . Furthermore, if H1 = H2 and K1 = K2 , then the map γ is skew-symmetric, in other words (xK1 , xK2 )γ = K3 for all x ∈ H1 . Proof This lemma is a slightly more general form of [11, Chapter VIII, Lemma 6.1] and can be proved in the same way. The last statement of the lemma is obvious. ¤ If Hi /Ki has exponent p for i = 1, 2, 3, where p is a prime, then each Hi /Ki can be regarded as a vector space over the field Fp , and the map γ is obviously Fp -bilinear.

14

CHAPTER 2. TECHNICAL RESULTS

A different formulation of Lemma 2.2.1 is that (with the given hypotheses) the map ϕy : H1 /K1 → H3 /K3 such that (xK1 )ϕy = [x, y]K3 is a well-defined group homomorphism for all y ∈ H2 , and the map γˆ : H2 /K2 → Hom(H1 /K1 , H3 /K3 ) such that (yK2 )γˆ = ϕy is also a well-defined group homomorphism. More generally we have the following result. Lemma 2.2.2 Assume the hypotheses of Lemma 2.2.1. In addition, suppose that Q is a group of operators for G and that Hi and Ki are Q-subgroups of G, whence in particular Hi /Ki becomes a ZQ-module (i = 1, 2, 3). Suppose also that Q centralizes H2 /K2 . Then the map ϕy : H1 /K1 → H3 /K3 such that (xK1 )ϕy = [x, y]K3 for all x ∈ H1 is well defined and a ZQ-module homomorphism for all y ∈ H2 , and the map γˆ : H2 /K2 → HomZQ (H1 /K1 , H3 /K3 ) such that (yK2 )γˆ = ϕy for all y ∈ H2 is well defined and a group homomorphism. Proof The fact that ϕy is a group homomorphism for all y ∈ H2 follows easily from Lemma 2.2.1. In order to show that the maps ϕy are actually ZQ-homomorphisms it is sufficient to notice that for all x ∈ H1 , y ∈ H2 and ξ ∈ Q we have ((xK1 )ϕy )ξ = [x, y]ξ K3 = [xξ , y ξ ]K3 = (xξ K1 , y ξ K2 )γ = (xξ K1 , yK2 )γ = (xξ K1 )ϕy . This proves that ϕy ∈ HomZQ (H1 /K1 , H3 /K3 ). It is an easy consequence of Lemma 2.2.1 that γˆ is a group homomorphism. ¤ Later on, namely in Section 3.4, we shall see how to handle also the case in which Q does not centralize H2 /K2 . For the moment, let us suppose that

2.2. LINEAR MAPS AND COMMUTATION

15

we are interested in obtaining a single homomorphism ϕw : H1 /K1 → H3 /K3 (defined as in Lemma 2.2.2) for a fixed w ∈ H2 . We may apply Lemma 2.2.2 with H2 = hwi and K2 = 1, but since now we do not care anymore about the linearity of ϕw with respect to w we may find the hypotheses of Lemma 2.2.2 too restrictive (for instance we may wish to replace [H1 , hwi] ≤ H3 with the weaker assumption [H1 , w] ≤ H3 ). In the following lemma we shall also drop the assumption that the factor groups H1 /K1 and H3 /K3 are abelian (notice that the subgroups H3 and K3 will be renumbered). Lemma 2.2.3 Suppose that H1 , H2 , K1 , K2 are subgroups of a group G, with Ki C Hi (i = 1, 2). Fix an element w of G and suppose that [H1 , w] ≤ H2 and that [K1 , w] and [H1 , w, H1 ] are both contained in K2 . Then the map ϕw : H1 /K1 → H2 /K2 such that (xK1 )ϕw = [x, w]K2 for all x ∈ H1 is well defined and a group homomorphism. Furthermore, if Q is a group of operators for G which centralizes w and if H1 , H2 , K1 , K2 are Q-subgroups of G, then the map ϕw is a Q-homomorphism. Proof Since [H1 , w] ≤ H2 , we have [x, w] ∈ H2 for all x ∈ H1 . The following commutator identity holds for all x, y ∈ G: [xy, w] = [x, w]y [y, w] = [x, w] [x, w, y] [y, w]. If x ∈ H1 and y ∈ K1 , then we have [x, w, y] ∈ K2 because [H1 , w, K1 ] ≤ K2 , and [y, w] ∈ K2 because [K1 , w] ≤ K2 ; hence [xy, w]K2 = [x, w]K2 , and this shows that the map ϕw is well defined. For x, y ∈ H1 we have [x, w, y] ∈ K2 since [H1 , w, H1 ] ≤ K2 , and this proves that ϕw is a group homomorphism. Now suppose that the group Q acts on G by automorphisms normalizing H1 , H2 , K1 , K2 and centralizing w. In particular Q acts on H1 /K1 and H2 /K2 . Then, for all x ∈ H1 and ξ ∈ Q, we have ((xK1 )ϕw )ξ = [x, w]ξ K2 = [xξ , wξ ]K2 = [xξ , w]K2 = (xξ K1 )ϕw . Thus ϕw is a Q-homomorphism.

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The special case of Lemma 2.2.3 in which K1 = K2 = 1 will be most useful; in this case, the hypothesis [K1 , w] ≤ K2 of Lemma 2.2.3 is trivially satisfied, and thus only the hypotheses [H1 , w] ≤ H2 and [H1 , w, H1 ] = 1 survive. If we also drop the hypothesis [H1 , w, H1 ] = 1, then the map ϕw is not a group homomorphism in general, but it satisfies the rule (xy)ϕw = (xϕw )y y ϕw for all x, y ∈ H1 . It is clearly possible to obtain a group homomorphism from ϕw by restricting ˆ 1 of H1 is a group its domain; in fact, the restriction of ϕw to any subgroup H ˆ 1 , w, H ˆ 1 ] = 1. Though there may not be in homomorphism exactly when [H ˆ 1 of G which is maximal among those which general a unique subgroup H ˆ 1 will be given in Lemma 2.2.4. satisfy this property, a sensible choice of H To proceed systematically, let us first remark that while the image of ϕw is not in general a subgroup of H2 , the inverse image of the trivial subgroup under ϕw is CH1 (w), and thus it is a subgroup of H1 , although not necessarily normal. (Notice also that CH1 (w) satisfies one of the properties of the kernel of a homomorphism, namely xϕw = y ϕw ⇐⇒ xy −1 ∈ CH1 (w), though ϕw is not a homomorphism in general.) ˆ 2 is a subgroup of H2 such that [H1 , H ˆ 2 ] = 1, and if More generally, if H ˆ ˆ ˆ H1 denotes the inverse image of H2 under ϕw , then H1 is a subgroup of H1 , ˆ 1 is a group homomorphism. Both assertions and the restriction of ϕw to H are straightforward consequences of the commutator formula [xy, w] = [x, w] [x, w, y] [y, w]. ˆ 1 we have that In fact, for x, y ∈ H ˆ 1 , w, H ˆ 1 ] ≤ [H ˆ 2 , H1 ] = 1; [x, w, y] ∈ [H ˆ 1 is a subgroup of G and the restriction of ϕw to H ˆ 1 is a group therefore H homomorphism. If, in addition, Q is a group of operators for G which centralizes w and ˆ 2 , then H ˆ 1 is clearly a Q-subgroup of G, and the restriction of normalizes H ˆ ϕw to H1 is a Q-homomorphism. What we have just proved is stated in the following lemma.

2.3. IRREDUCIBLE MODULES FOR ABELIAN GROUPS

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Lemma 2.2.4 Let H1 , H2 be Q-subgroups of the Q-group G (with Q possibly the trivial group), and let w be an element of G which is centralized by Q ˆ 2 = CH2 (H1 ), and and such that [H1 , w] ≤ H2 . Let us put H ˆ 1 = {h ∈ H1 | [h, w] ∈ H ˆ 2 }. H ˆ 1 is a Q-subgroup of H1 , the map Then H ˆ1 → H ˆ2 ϕw : H x 7→ [x, w] is a Q-homomorphism, and the kernel of ϕw is CHˆ 1 (w) = CH1 (w).

2.3

Irreducible modules for abelian groups

The following theorem describes all irreducible modules for a finite abelian group over a finite field. With an abuse of language, by a faithful FG-module we shall always mean an FG-module which is faithful for G (that is to say, no non-identity element of G acts trivially on it), but not necessarily for the group algebra FG. Theorem 2.3.1 Let Fpf be the finite field of order pf , let A be an abelian group and let V be a faithful irreducible Fpf A-module of dimension n over Fpf . Then: (i) A is cyclic of order prime to p; (ii) n is the smallest positive integer such that |A| divides (pnf − 1) (we also say that n is the multiplicative order of pf modulo |A|); thus Fpnf is the smallest extension field of Fpf which contains a primitive |A|th root of unity ε, in particular we have Fpf (ε) = Fpnf ; (iii) if a0 is a generator of A, then there exists a primitive |A|th root of unity ε in Fpnf such that V is isomorphic to the Fpf A-module Vε whose underlying vector space over Fpf is the field Fpnf and where the action of A on Vε is given by xai0 = εi x for all x ∈ Fpnf and for all i = 0, . . . , |A| − 1.

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(iv) the ring EndFpf A (V ) is a field isomorphic to Fpnf , and consists of the maps ϕy : V

→ V

v 7→ vy, for y ∈ Fpf A. Proof Statements (i), (ii), and (iii) of the theorem follow from [10, Kapitel II, Satz 3.10] and its proof. Let us observe that [10, Kapitel II, Satz 3.10] only states that the actions of A on V and on Vε are permutation isomorphic, but it is clear from its proof that V and Vε are actually isomorphic as Fpf Amodules. In order to prove assertion (iv) of the theorem, we may take V = Vε , according to statement (iii). Then each of the maps ϕy : Vε → Vε is the multiplication by some fixed element of Fpnf , and thus it is clearly an endomorphism of Vε as an Fpf A-module. Hence the set of the maps ϕy for y ∈ Fpf A is a field isomorphic to Fpnf , and is a subring of EndFpf A (Vε ). According to Schur’s Lemma EndFpf A (Vε ) is a division ring; therefore, Vε is a vector space over EndFpf A (Vε ). From the fact that Vε has order pnf it follows now that Vε has dimension 1 over EndFpf A (Vε ), and that EndFpf A (Vε ) = {ϕy | y ∈ Fpf A}. The proof is complete.

¤

We observe that the choice of the primitive |A|th root of unity ε in statement (iii) is not arbitrary. In fact, different choices of ε may give rise to non-isomorphic Fpf A-module structures on Fpnf . More precisely, we shall see that Vε and Vε0 are isomorphic if and only if ε and ε0 , which are primitive |A|th roots of unity in Fpnf , are Galois conjugate over Fpf . Let m(x) ∈ Fpf [x] be the minimal polynomial of the Fpf -linear transformation induced by a0 on V (via the module action). According to Theorem 2.3.1, our m(x) is also the minimal polynomial of the Fpf -linear transformation of Fpnf given by multiplication by ε. It follows that m(x) is the minimal polynomial of ε over Fpf , in particular ε is an eigenvalue of a0 on V .


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Furthermore, the eigenvalues of a0 on V are exactly the Galois conjugates f

2f

(n−1)f

ε, εp , εp , . . . , εp

.

In fact, since ε is a root of m(x), and because the coefficients of m(x) belong to the field Fpf , we have if

if

m(εp ) = m(ε)p = 0 for all i = 0, . . . , n − 1; f

2f

(n−1)f

hence ε, εp , εp , . . . , εp are roots of m(x). They are pairwise distinct as Fpf (ε) = Fpnf has dimension n over Fpf . Since the degree of m(x) does not exceed the dimension n of V , they are exactly all the roots of m(x), that is to say, all the eigenvalues of a0 on V . Now let us extend the ground field Fpf of V to E = Fpnf . The tensor product V E = V ⊗Fpf E is a vector space over E and becomes an EA-module in a natural way (see [10, Kapitel V, Hilfsatz 11.1 and Hilfsatz 11.3]). The E-linear transformation induced by a0 on V E can be put into diagonal form, because it has all its eigenvalues in E, and they are all distinct. Hence there exist an E-basis v0 , . . . , vn−1 of V E such that if

vi a0 = εp vi for all i = 0, . . . , n − 1. We have seen that an irreducible faithful module for a cyclic group over a finite field Fpf determines a Galois conjugacy class of roots of unity over Fpf . The following result is stronger. Theorem 2.3.2 Let A = ha0 i be a cyclic group and let E be a splitting field for the polynomial x|A| − 1 over Fpf . The isomorphism classes of irreducible Fpf A-modules are in a bijective correspondence with the orbits of |A|th roots of unity in E under the Galois group of E over Fpf . This correspondence associates to each irreducible Fpf A-module V the set of the eigenvalues of a0 on V . Proof Let us first prove the theorem under the assumption that |A| is not divisible by p. Let R be the set of the |A|th roots of unity in E. Since E is a splitting field for x|A| − 1, and since p does not divide |A|, the field E contains |A| distinct |A|th roots of unity, and therefore |R| = |A|. If ε ∈ R, the extension

20


field Fpf (ε) can be regarded as a vector space over Fpf and becomes an Fpf Amodule if one defines vai0 = εi v for all v ∈ Fpf (ε) and for all i = 1, . . . , |A|. This Fpf A-module, which we shall denote by Vε , is irreducible, because any Fpf A-submodule of Vε is an ideal of the field Fpf (ε), and hence is either 0 or Vε . If V is any irreducible Fpf A-module, then V can be regarded as a faithful irreducible module for A¯ = A/K over Fpf , where K = {a ∈ A | va = v for all v ∈ V } is the kernel of the representation of A on V . It follows from statement ¯ (iii) of Theorem 2.3.1 that there exists a primitive |A|th root of unity ε (in particular ε is an |A|th root of unity and we may assume that ε ∈ R) such ¯ that V is isomorphic to Vε as an Fpf A-module, and hence as an Fpf A-module. Thus the set of modules Vε for ε ∈ R contains a complete set of representatives for the isomorphism classes of irreducible Fpf A-modules. As we have seen above, the eigenvalues of a0 on Vε are exactly all the distinct Galois conjugates of ε. In particular if Vε and Vε0 are isomorphic (for ε, ε0 ∈ R), then ε and ε0 are Galois conjugate, in other words they belong to the same orbit of the Galois group of E over Fpf on R. Conversely, assume now that ε and ε0 are Galois conjugate. Then since f the Galois group of E over Fpf is generated by the automorphism c 7→ cp , if we have ε0 = εp for some integer i. In particular Vε and Vε0 have the same underlying vector space over Fpf , namely the field Fpf (ε) = Fpf (ε0 ). The map if θ : Vε → Vε0 such that v θ = v p is then an isomorphism of Fpf A-modules, because it is an isomorphism of vector spaces over Fpf and it satisfies if

if

if

(va0 )θ = (vε)p = v p εp = v θ a0 for all v ∈ Vε . Hence Vε and Vε0 are isomorphic if and only if ε and ε0 are Galois conjugate. Thus the theorem is proved under the additional assumption that |A| is not divisible by p. Now let us drop this assumption, and let pr be the highest |A|/pr power of p which divides |A|. Let us put A¯ = A/P , where P = ha0 i is the Sylow p-subgroup of A.


21

According to [10, Kapitel V, Satz 5.17] P is contained in the kernel of every irreducible representation of A over Fpf ; hence each irreducible Fpf A¯ module V can be regarded as an irreducible Fpf A-module. On the other hand, since r ¯ r ¯ x|A| − 1 = x|A|p − 1 = (x|A| − 1)p , ¯

the splitting field E for x|A| − 1 over Fpf is also a splitting field for x|A| − 1 ¯ and the set R of |A|th roots of unity in E coincides with the set of |A|th roots of unity in E. ¯ the theorem is true for the group A, ¯ as we Since p does not divide |A|, have proved above. Thus, in view of these observations, its conclusion holds for the group A too. ¤ We observe here that Theorem 2.3.2 in disguise says that the isomorphism classes of irreducible Fpf A-modules are in a bijective correspondence with the Galois conjugacy classes over Fpf of irreducible E-characters of A; thus Theorem 2.3.2 may also be deduced from [13, Theorem 9.21] together with [13, Corollary 9.7]. An important consequence of Theorem 2.3.2 is that the isomorphism class of an irreducible Fpf A-module V (for a cyclic group A = ha0 i) is uniquely determined by the isomorphism class of the (not necessarily irreducible) EAmodule V E . In fact, we saw that V E has an E-basis whose elements are eigenvectors for a0 and, according to Theorem 2.3.1, the corresponding eigenvalues form an orbit under the Galois group of E over Fpf and determine the Fpf A-module V up to isomorphism. We shall need the following more general result. Corollary 2.3.3 Let W be a semisimple module for the cyclic group A over Fpf and let E be a splitting field for the polynomial x|A| − 1 over Fpf . Then the isomorphism class of W as an Fpf A-module is uniquely determined by the isomorphism class of W E as an EA-module. L

Proof Since W is semisimple, we have W = ri=1 Vi , where the Vi are irreL ducible Fpf A-modules. It is easy to see that W E ∼ = ri=1 ViE . Since we saw that ViE determines Vi uniquely up to isomorphism, the desired conclusion follows. ¤

22


Let us notice that Corollary 2.3.3 remains true if we replace the cyclic group A with an arbitrary finite group. This is again a consequence of [13, Theorem 9.21 and Corollary 9.7].

2.4. HOMOGENEOUS MODULES

2.4

23

Homogeneous modules

An FQ-module V (where F is a field and Q is a group) is said to be homogeneous if it is semisimple and all its FQ-composition factors are isomorphic. If V is a semisimple FQ-module (for instance V is always semisimple when the characteristic of F is 0 or a prime not dividing the order of Q, according to Maschke’s Theorem), then by definition V is a direct sum of irreducible FQ-submodules. A submodule of V which is the sum of all irreducible submodules isomorphic to a fixed irreducible FQ-module is called a homogeneous component of V . It is easy to see that each homogeneous component of V is invariant under FQ-endomorphisms of V , and that V is the direct sum of its homogeneous components (see for instance [13, Lemma 1.13]). Lemma 2.4.1 Let F be a field, Q an abelian group and S a semisimple FQ-module. Then the following assertions are equivalent: (i) every cyclic FQ-submodule of S is irreducible; (ii) S is the union of its irreducible FQ-submodules; (iii) S is FQ-homogeneous. Proof ((i)⇒ (ii)) If each element of S generates an irreducible submodule of S, then S is clearly a union of irreducible submodules. ((ii)⇒ (iii)) Since S is semisimple, S is the direct sum of its FQhomogeneous components and every irreducible FQ-submodule of S is contained in some FQ-homogeneous component of S. Hence S has only one FQ-homogeneous component, that is to say, S is FQ-homogeneous. ((iii)⇒ (i)) Since S is semisimple, S can be written as an internal direct sum S = V1 ⊕ · · · ⊕ Vk of irreducible FQ-submodules V1 , . . . , Vk , which are all isomorphic because S is FQ-homogeneous. We may therefore assume that S = V ⊕ · · · ⊕ V , the external direct sum of k isomorphic copies of an irreducible FQ-module V . Let (v1 , . . . , vk ) be a non-zero element of S. We may assume v1 6= 0. Since V is irreducible we have v1 FQ = V , in particular for each i = 2, . . . , k there exists an element ai of FQ such that v1 ai = vi .

24

CHAPTER 2. TECHNICAL RESULTS Now the map ϕ:V

→ S

v 7→ (v, va2 , . . . , vak ) is clearly F-linear. Furthermore, the map ϕ is an FQ-homomorphism, because Q is abelian. Since V is irreducible and ϕ is not the zero homomorphism, ϕ is a monomorphism. Thus its image (v1 , . . . , vk )FQ, namely the cyclic FQsubmodule of S generated by (v1 , . . . , vk ), is isomorphic to V , in particular it is an irreducible FQ-module. ¤ The notion of homogeneous component generalizes to a situation in which the modules are not semisimple, namely that of abelian groups (or, in other words, Z-modules) with operator groups of coprime order. To proceed systematically, we shall first recall some well-known facts. Theorem 2.4.2 Let A be an abelian p-group, and let Q be a p0 -group of automorphisms of A. Suppose that A is indecomposable as a Q-group. Then A is homocyclic (in other words, A is the direct product of cyclic groups of the same order), and the only Q-subgroups of A are i

Ωi (A) = {a ∈ A | ap = 1}, for i = 0, 1, . . . , e, where pe is the exponent of A. Furthermore, all Qcomposition factors of A are Q-isomorphic. Proof See [11, Chapter VIII, Theorem 5.9 and Theorem 5.10]. The fact that all Q-composition factors of A are Q-isomorphic is a consequence of the fact that the map A → A a 7→ ap is a Q-endomorphism of A.

¤

Now let A be an abelian p group with a p0 -group of automorphisms Q. If V is a fixed irreducible Fp Q-module, let B be the product of all indecomposable Q-subgroups of A which have some Q-composition factor isomorphic

2.5. INDUCTION AND TENSOR INDUCTION

25

to V (as a Q-group, or equivalently as an Fp Q-module). It is an easy consequence of Theorem 2.4.2 that all Q-composition factors of B are Q-isomorphic to V . We shall thus call B a Q-homogeneous component of A. The name is justified by the easy facts that Q-homogeneous components of A are invariant under Q-endomorphisms of A, and that A is the direct product of its Q-homogeneous components. (We observe that the statement of Lemma 2.1.1 which refers to A abelian is a special case of this, CQ (A) being the Q-homogeneous component of A which corresponds to the trivial module.)

2.5

Induction and tensor induction

While the reader is certainly familiar with induction of modules and characters, he may not be so with tensor induction. This technique is particularly useful for the description of the representations of wreath products. In order to show the similarity between ordinary induction and tensor induction we shall give a brief exposition of both techniques in succession. Expositions of tensor induction can also be found in [5, §13] and [14, Section 4]. Let H be a subgroup of the group G, let F be a field and W a right FHmodule. Since the group algebra FG is an (FH, FG)-bimodule, the tensor product W ⊗FH FG becomes a right FG-module according to [10, Kapitel V, Satz 9.8], which is called the induced module and is denoted by W G . L Let T be a right transversal for H in G. Then FH = t∈T (FH)t is a decomposition of FG as a left FH-module (which shows that FG is a free left FH-module). Hence we have the decomposition W G = W ⊗FH FG =

M

(W ⊗ t)

t∈T

of W G as a vector space over F, where W ⊗ t = {w ⊗ t | w ∈ W }. Each W ⊗ t is an FH t -submodule of W G . In fact, for h ∈ H we have (w ⊗ t)ht = w ⊗ ht = wh ⊗ t. Now G acts on the set of right cosets of H in G by right multiplication, hence it acts on T . Let us denote this action by (t, g) 7→ t · g. In other words, t · g (for t ∈ T and g ∈ G) is the unique element of T such that tg ∈ H(t · g), or equivalently tg(t · g)−1 ∈ H.

26

CHAPTER 2. TECHNICAL RESULTS We therefore have (w ⊗ t)g = wtg(t · g)−1 ⊗ (t · g) ∈ W ⊗ (t · g),

where we observe that in order to compute wtg(t · g)−1 it is sufficient to know W as an FH-module, because tg(t · g)−1 ∈ H. More generally, any element P of W G has the form t∈T (wt ⊗ t) for some wt ∈ W , and we have Ã

X

!

X

(wt ⊗ t) g =

t∈T

(xt ⊗ t),

t∈T

where xt = wt·g−1 ((t · g −1 )gt−1 ). There is an alternative definition of W G which builds it as the direct sum of F-spaces W ⊗ t, each isomorphic to W via the map w 7→ w ⊗ t, and makes it into an FG-module according to the formula above. Although our original definition of W G is to be preferred as it does not involve any choice of a transversal T , the formula above will serve as a model for the definition of the tensor induced module W ⊗G . As we have seen, W G is the direct sum of the F-subspaces W ⊗t for t ∈ T , and each W ⊗ t is an FH t -submodule of W G . Let us define the F-space W ⊗G =

O

(W ⊗ t),

t∈T

namely W ⊗G is the tensor product over F of the F-spaces W ⊗ t, where we are assuming some fixed but arbitrary total order on T . The FH t -module structure of each W ⊗ t can now be used to give W ⊗G an FG-module structure, much in the same way as for W G . We define an N action of g ∈ G on the pure tensors t∈T (wt ⊗ t) (for some wt ∈ W ), as follows: Ã ! O

O

t∈T

t∈T

(wt ⊗ t) g =

(xt ⊗ t),

where xt = wt·g−1 ((t·g −1 )gt−1 ). We observe that this formula can be obtained P N from that which gives the action of g on W G by replacing t∈T with t∈T . Consequently, the action of g on the pure tensors of W ⊗G can be described in terms of the action of g on W G as follows: Ã O

!

(wt ⊗ t) g =

t∈T

O t∈T

ÃÃ

X

s∈T

! !πt

(ws ⊗ s) g

,


27

where the maps πt : W G → W ⊗ t are the projections on the summands of L W G = t∈T (W ⊗ t). Since the action of g on the pure tensors is linear in each wt ⊗t, it extends to a unique and well-defined action of g on W ⊗G . It is easy to check that ÃÃ O

!

!

(wt ⊗ t) g1 g2 =

O

(wt ⊗ t)g1 g2 ;

t∈T

t∈T

hence W ⊗G becomes an FG-module. We observe that dim W G = |G : H| dim W, and that dim W ⊗G = (dim W )|G:H| . It is not difficult to show that the isomorphism class of W ⊗G depends only on the isomorphism class of W , and not on the transversal T , or on the particular ordering given to it. Now let ψ be the character of H afforded by W and let ψ G and ψ ⊗G be the characters of G afforded by W G and W ⊗G respectively. We shall compute explicit expressions for ψ G and ψ ⊗G in terms of ψ. Lemma 2.5.1 Let g ∈ G and let ψ ◦ denote the function from the group G to the field F which coincides with ψ on the subgroup H and takes the value zero on G \ H. Then we have ψ G (g) =

X

ψ ◦ (tgt−1 ).

t∈T

Proof Let w1 , . . . , ws be a basis of W . Then a basis of W G is given by the elements wi ⊗ t for i = 1, . . . s and for t ∈ T . Let us write wi h =

s X

aij (h)wj

j=i

for h ∈ H, with aij (h) ∈ F. In particular, we have ψ(h) = the other hand, we have (wi ⊗ t)g = wi (tg(t · g)−1 ) ⊗ (t · g) =

s X j=1

Ps

i=1

aii (h). On

aij (tg(t · g)−1 )wj ⊗ (t · g).

28


Now we have wi ⊗ t = wj ⊗ (t · g) exactly when i = j and t = t · g, or equivalently when i = j and tgt−1 ∈ H. It follows that ψ G (g) =

s XX

aii (tgt−1 ) =

t∈T 0 i=1

X

ψ ◦ (tgt−1 ),

t∈T

where T 0 is the set of the elements t of T such that tgt−1 ∈ H. The proof is complete. ¤ We observe that since ψ is a class function, when F has characteristic zero we can also write ψ G (g) =

1 X ◦ ψ (xgx−1 ). |H| x∈G

If we choose a set of representatives x1 , . . . , xm for the conjugacy classes of ˙ ˙ H H contained in g G (that is to say, g G ∩ H = xH 1 ∪ · · · ∪xm ), we obtain m |G : H| X |xH |ψ(xi ). ψ (g) = G |g | i=1 i G

The following useful formula follows: ψ G (g) = |CG (g)|

m X

ψ(xi ) . i=1 |CH (xi )|

This can also be expressed by saying that ψ G (g) equals |G : H| times the mean value of ψ ◦ over g G . Now let us pass to the computation of the tensor induced character ψ ⊗G . Lemma 2.5.2 Let g ∈ G and let T0 be a set of representatives for the orbits of hgi in its action on T via ·. For t ∈ T , let n(t) denote the size of the hgi-orbit which contains t. Then we have ψ ⊗G (g) =

Y

ψ(tg n(t) t−1 ).

t∈T0

Proof Let Ω1 , . . . , Ωr be the orbits of hgi on T , let us choose a set T0 = {t1 , . . . , tr }


29

of representatives for them (with ti ∈ Ωi ), and let us put n(i) = |Ωi |. Hence hgi/hg n(i) i acts regularly on Ωi and we have Ωi = {ti · g, . . . , ti · g n(i) = ti }. Since the isomorphism class of W ⊗G is independent of the ordering given to T , we are allowed to order T as follows: T = {t1 · g, . . . , t1 · g n(1) , . . . , tr · g, . . . , tr · g n(r) }. Now W ⊗G regarded as an Fhgi-module is isomorphic to the tensor product N module ri=1 Wi , where Wi is the Fhgi-module n(i) O

(W ⊗ (ti · g j )).

Wi =

j=1

If ψi denotes the character of hgi afforded by Wi , we have ψ

⊗G

(g) =

r Y

ψi (g),

i=1

according to [13, Theorem 4.1]. Let us fix an index i = 1, . . . , r. We shall prove that ψi (g) = ψ(ti g n(i) t−1 i ). Let w1 , . . . , ws be an F-basis of W , then an F-basis of Wi is given by the tensors n(i) O

(wkj ⊗ (ti · g j )),

j=1

for (k1 , . . . , kn(i) ) ∈ {1, . . . , s}n(i) . The action of g on any element of this basis of W ⊗G is given by the formula  

n(i) O

 j

(wkj ⊗ (ti · g )) g =

j=1

n(i) O

(xj ⊗ (ti · g j )),

j=1

where xj = wkj−1 ((ti · g j−1 )g(ti · g j )−1 ), and the index j is read mod n(i) (in particular, wk0 = wkn(i) ). Let Aj = (ajkl )k,l=1,... ,s be the matrix of the action of ((ti · g j−1 )g(ti · g j )−1 ) on W with respect to the basis w1 , . . . , ws ; in other words s wk ((ti · g j−1 )g(ti · g j )−1 ) =

X l=1

ajkl wl .

30

CHAPTER 2. TECHNICAL RESULTS Nn(i)

Now the coefficient of the basis element j=1 (wkj ⊗ (ti · g j )) in the expression Nn(i) of j=1 (xj ⊗ (ti · g j )) as a linear combination of the elements of the given basis of Wi is n(i)

Y

aj,kj−1 ,kj .

j=1

Hence we have ψi (g) =

X n(i) Y

aj,kj−1 ,kj = tr(A1 A2 · · · An(i) ),

j=1

where the sum is taken over (k1 , . . . , kn(i) ) ∈ {1, . . . , s}n(i) . The matrix A1 A2 · · · An(i) is the matrix of the action of n(i)

Y

((ti · g j−1 )g(ti · g j )−1 ) = ti g n(i) t−1 i

j=1

on W , with respect to the basis w1 , . . . , ws . Therefore we have ψi (g) = ψ(ti g n(i) t−1 i ), as claimed. It follows that ψ ⊗G =

r Y

ψ(ti g n(i) t−1 i ),

i=1

which concludes the proof.

2.6

¤

Basic commutators

In Chapter 6 we shall need the notion of basic commutators. Definition 2.6.1 [8, p. 178] Let F be a free group of rank n, generated by x1 , . . . , xn . The basic commutators on x1 , . . . , xn are the elements of the ordered infinite set {ci }i∈N , defined inductively as follows: (i) ci = xi , for i ≤ n, are the basic commutators of weight 1, and are ordered by the rule c1 < c2 < · · · < cn ;

2.6. BASIC COMMUTATORS

31

(ii) if basic commutators of weight less than l have been defined and ordered, then the commutator [u, v] is a basic commutator exactly when (a) u and v are basic commutators, and the sum of their weights is l, (b) u > v, (c) if u = [w, t], then v ≥ t; furthermore, commutators of weight l follow all those of lower weight, and if [u1 , v1 ] and [u2 , v2 ] have weight l, then we have [u1 , v1 ] < [u2 , v2 ] if either v1 < v2 or v1 = v2 and u1 < u2 . We state without proof the following theorem (see [8, Theorem 11.2.4] for a proof). Theorem 2.6.2 If F is a free group with free generators x1 , . . . , xn , and if l ≥ 1, then an arbitrary element f of F has a unique representation f ≡ ce11 ce22 · · · cet t mod γl+1 (F ), where c1 , . . . , ct are the ordered basic commutators of weight less than or equal to n, and e1 , . . . , et are integers. In particular, for all l ≥ 1, the factor group γl (F )/γl+1 (F ) is a free abelian group (this is also in [11, Chapter VIII, Theorem 11.15]), and the basic commutators of weight l form (a set of representatives of ) a basis for γl (F )/γl+1 (F ) over Z. The analysis of the descending central series of a free group F in terms of basic commutators, provided by Theorem 2.6.2, will be used in Chapters 5 and 6 for the construction of certain groups of exponent p, where p is a prime. Thus we shall be mainly interested in the structure of the descending central series of the factor group F¯ = F/F p (of course this is a hard problem in general, directly related to the famous Burnside’s problem, see [10, Kapitel III, Bemerkungen 6.7]). The factors γl (F¯ )/γl+1 (F¯ ) of the descending central series of F¯ are finite elementary abelian p-groups; hence, they can be regarded as vector spaces over Fp (while the factors γl (F )/γl+1 (F ) can be regarded as free Zmodules). It is easy to see that the basic commutators of weight l generate

32


γl (F¯ )/γl+1 (F¯ ). Unfortunately, what would be the analogue for F¯ of the last statement of Theorem 2.6.2 does not hold in general: the basic commutators of weight l in F¯ do not constitute a basis of γl (F¯ )/γl+1 (F¯ ) over Fp in general. For instance, when p = 2 we have F 0 ≤ F 2 (see [10, Kapitel III, Satz 3.14]); consequently, F¯ is an elementary abelian 2-group, and thus all commutators of weight l > 1 are trivial. However, the last statement of Theorem 2.6.2 passes on to F¯ (with the word Z-basis replaced by Fp -basis) for l < p. Let us state this fact as a theorem. Theorem 2.6.3 Let F be a free group with free generators x1 , . . . , xn , let p be a prime, and let F¯ = F/F p . Then, for all l ≥ 1, the factor group γl (F¯ )/γl+1 (F¯ ) is a finite elementary abelian p-group, and it is generated by the basic commutators of weight l. Furthermore, if l < p, the basic commutators of weight l form a basis of γl (F¯ )/γl+1 (F¯ ) over Fp . Theorem 2.6.3 is an easy consequence of a well-known result (see for instance [21, Lemmas 1.11 and 1.12]), which gives Fp -bases, in terms of basic commutators and their powers, for the factors of the p-lower central series κl (F ) of F (which is defined in [11, Chapter VIII, Definition 1.10] as κl (G) =

Y

k

γi (G)p ,

ipk ≥l

for an arbitrary group G, and with respect to a fixed prime p): κl (F )/κl+1 (F ) is an elementary abelian p-group, and (a set of representatives of) a basis for k it over Fp is given by the set of all elements of F of the form cp , where pk divides l, and c is some basic commutator of weight l/pk (in particular, the basic commutators of weight l form a basis of κl (F )/κl+1 (F ) over Fp for l not a multiple of p). Now since κl (F ) = F p γl (F ) for l ≤ p, we have κl (F¯ ) = γl (F¯ ) for l ≤ p, and Theorem 2.6.3 follows. However, we shall give here a more self-contained proof of Theorem 2.6.3, after recalling without proof the following lemma. Lemma 2.6.4 If x, y are elements of a group G, and p is a prime, then (xy)p ≡ xp y p mod γ2 (G)p γp (G)

2.6. BASIC COMMUTATORS

33

(that is to say, the map G → G/γ2 (G)p γp (G) x 7→ xp γ2 (G)p γp (G) is a group homomorphism). Proof This is a special case of [11, Chapter VIII, Lemma 1.1].

¤

Proof of Theorem 2.6.3 We have γl (F¯ ) = γl (F )F p /F p for l ≥ 1, and thus γl (F¯ )/γl+1 (F¯ ) = γl (F )F p /γl+1 (F )F p ∼ = γl (F )/γl (F ) ∩ γl+1 (F )F p = γl (F )/(γl (F ) ∩ F p )γl+1 (F ). Hence γl (F¯ )/γl+1 (F¯ ) is isomorphic to a factor group of the free abelian group γl (F )/γl+1 (F ). It follows from Theorem 2.6.2 that the basic commutators of weight l on x1 , . . . , xn (as elements of F¯ ) generate γl (F¯ )/γl+1 (F¯ ). Consequently, and because F¯ has exponent p, the factor group γl (F¯ )/γl+1 (F¯ ) is a finite elementary abelian p-group, and its dimension over Fp is at most the rank of the free abelian group γl (F )/γl+1 (F ). Now the basic commutators of weight l on x1 , . . . , xn clearly form an Fp basis of γl (F )/γl (F )p γl+1 (F ). Consequently, the conclusion of the theorem will follow if we can prove that (γl (F ) ∩ F p )γl+1 (F ) = γl (F )p γl+1 (F ) for l < p. Since the inclusion ≥ is clearly true for all l, we only have to prove that, for l < p, γl (F ) ∩ F p ≤ γl (F )p γl+1 (F ). We shall prove the following equivalent statement, which lends itself to an inductive argument: γl (F ) ∩ γl−k (F )p ≤ γl (F )p γl+1 (F ) for all k = 0, . . . , l. This is certainly true for k = 0. Now let 0 < k < l, and let assume that γl (F ) ∩ γl−k+1 (F )p ≤ γl (F )p γl+1 (F )

34


has already been proved. Let g be an element of γl (F ) ∩ γl−k (F )p ; then g can be written as a product g=

r Y p

yj ,

j=1

with y1 , . . . , yr ∈ γl−k (F ). According to Lemma 2.6.4, we have that g=



r Y p

yj ≡ 

j=1

p

r Y

yj  mod γ2 (G)p γp (G),

j=1

where we have put G = γl−k (F ). In particular, the above congruence holds modγl−k+1 (F )p γp (F ), because according to [10, Kapitel III, Hauptsatz 2.11 b)] we have that γ2 (γl−k (F )) ≤ γ2(l−k) (F ) ≤ γl−k+1 (F ), and clearly γp (γl−k (F )) ≤ γp (F ). Q Now we shall prove that rj=1 yj ∈ γl−k+1 (F ). According to Theorem 2.6.2, we can write r Y

yj ≡

j=1

t Y ei

ci mod γl−k+1 (F ),

i=s

where cs , . . . , ct are the basic commutators of weight l − k, and es , . . . , et are integers. As before, Lemma 2.6.4 yields that Ã t !p Y e i

ci

i=s

≡

t Y pei

ci

mod γ2 (G)p γp (G),

i=s

where G = γl−k (F ); in particular this holds modγl−k+1 (F ) (let us notice that here we are not using the fact that l < p, because γp (γl−k (F )) ≤ γp(l−k) (F ) ≤ γl−k+1 (F ), by a repeated application of [10, Kapitel III, Hauptsatz 2.11 b)]). Now we obtain that p  g≡

r Y

j=1

yj  ≡

t Y pei

ci

i=s

mod γl−k+1 (F ).

2.7. CHARACTER TABLE ISOMORPHISMS

35

On the other hand, g ∈ γl (F ) ≤ γl−k+1 (F ). According to Theorem 2.6.2, c1 , . . . , ct are Z-linearly independent in γl−k (F )/γl−k+1 (F ); consequently, we have es = · · · = et = 0, and thus r Y

yj ∈ γl−k+1 (F ),

j=1

as claimed. Since we found earlier that 

g≡

r Y

p

yj  mod γl−k+1 (F )p γp (F ),

j=1

it follows that g ∈ γl−k+1 (F )p γp (F ). Now our hypothesis that l < p comes into play, and yields that γp (F ) ≤ γl+1 (F ). Consequently, g ∈ γl (F ) ∩ γl−k+1 (F )p γl+1 (F ) = (γl (F ) ∩ γl−k+1 (F )p )γl+1 (F ). By inductive hypothesis, we finally obtain that g ∈ γl (F )p γl+1 (F ). This concludes the proof.

2.7

¤

Character table isomorphisms

As promised in Chapter 1, here is a more handy definition of ‘having identical character tables’. Definition 2.7.1 Let G1 , G2 be finite groups. We will say that G1 and G2 have identical character tables if there exist bijections α : G1 → G2 and β : Irr(G1 ) → Irr(G2 ), such that χβ (g α ) = χ(g) for all g ∈ G1 and for all χ ∈ Irr(G1 ). We shall also say that (α, β) is a character table isomorphism from G to H.

36


Since the irreducible characters of Gi (for i = 1, 2) form a basis for the space of class functions on Gi taking values in the field of complex numbers, if such α, β exist, α must send any conjugacy class of G1 onto a conjugacy class of G2 . It is also clear that α sends the identity class of G1 to the identity class of G2 , and that β sends the trivial character of G1 to the trivial character of G2 .

Chapter 3 Looking for a counterexample 3.1

Introduction

The work of this thesis began as an attempt to prove that the character table of a soluble group G determines the derived length of G. Our exposition will follow this approach, and thus this chapter begins with the following conjecture, which will eventually turn out to be false. Conjecture 3.1.1 Let G and H be groups with identical character tables and assume that G is metabelian. Then H is metabelian. We shall disprove this conjecture by exhibiting a counterexample. But what could such a counterexample look like? Let us choose a counterexample (G, H) to Conjecture 3.1.1 with |G| minimal and let (α, β) be a character table isomorphism from G to H. Then H 00 is the unique minimal normal subgroup of H. In fact, if this were not true, H would have some ¯ with H 00 6≤ K. ¯ But then the factor groups non-trivial normal subgroup K −1 α ¯ ¯ G/K, where K = K , and H/K would have identical character tables, and ¯ 00 = H 00 K/ ¯ K ¯ 6= 1; hence (G/K, H/K) ¯ would be a counterexample to (H/K) our conjecture with |G/K| < |G|, a contradiction. Thus H 00 is the unique minimal normal subgroup of H. Since G and H have isomorphic lattices of −1 normal subgroups, the subgroup (H 00 )α , which we will henceforth call N , is the unique minimal normal subgroup of G. Hence N and H 00 are elementary abelian p-groups for some prime p, and of course they are isomorphic as they have the same order (because α is a 37

38

CHAPTER 3. LOOKING FOR A COUNTEREXAMPLE

bijection). Throughout this chapter we shall assume that G and H are not p-groups. More precisely, we shall consider the following hypotheses. Hypotheses 3.1.2 Let G and H be groups with identical character tables via the bijections (α, β). Let us assume that G and H are not nilpotent, that G is metabelian and that H 00 is the unique minimal normal subgroup of H. −1 Thus N = (H 00 )α is the unique minimal normal subgroup of G, and in particular N and H 00 are elementary abelian p-groups for some prime p. Under these hypotheses we shall obtain in the next section a nice description of a minimal counterexample (G, H). Subsequently, in Chapter 6, we shall also construct a counterexample (G, H) with G and H nilpotent.

3.2

Structure of a minimal counterexample

Lemma 3.2.1 Assume Hypotheses 3.1.2. Then (i) G0 and H 0 are p-groups; ¯ ×Q ¯ in (ii) G0 has a complement W × Q in G and H 0 has a complement W 0 ¯ ¯ H, where W , W are (abelian) p-groups and Q, Q are cyclic p -groups; ¯ acts regularly on H 0 ; (iii) Q acts regularly on G0 and Q ¯ acts faith(iv) Q acts faithfully and irreducibly on N by conjugation and Q fully and irreducibly on H 00 by conjugation; (v) CW (G0 ) = 1 and CW¯ (H 0 ) = 1. Proof Since the Fitting subgroup F(G) of G is nilpotent, each Sylow subgroup of F(G) is normal in F(G) and hence in G. But G has a unique minimal normal subgroup N , which is a p-group; hence F(G) is a p-group. We have G0 ≤ F(G). Let P be a Sylow p-subgroup of G. Then from 0 G ≤ F(G) ≤ P it follows that P C G and hence F(G) = P . Let us put P¯ = P α . Because α is a bijection, P¯ is a Sylow p-subgroup of H and of course H 0 ≤ P¯ C H. Thus both G and H have a normal Sylow p-subgroup, which contains the derived subgroup, and assertion (i) is proved. ¯ be complements for P in G and for P¯ in H respectively; these Let Q and Q ¯ are exist according to the Theorem of Schur-Zassenhaus. Clearly Q and Q

3.2. STRUCTURE OF A MINIMAL COUNTEREXAMPLE

39

abelian. They are also non-trivial, because we assumed that G and H are not nilpotent. Since G and H are soluble groups, we have CG (F(G)) ≤ F(G) and CH (F(H)) ≤ F(H) (see [10, Kapitel III, Satz 4.2]); in particular it follows that CQ (P ) = 1 and CQ¯ (P¯ ) = 1. Now we shall prove that Q acts regularly on G0 . Let K be a non-trivial subgroup of Q. Then, according to Lemma 2.1.1, we have G0 = [G0 , K] × CG0 (K). Since [G0 , K] = [G0 , G0 K] and CG0 (K) = CG0 (G0 K) are normal subgroups of G, and since G has the unique minimal normal subgroup N , either [G0 , K 0 ] or CG0 (K) is trivial. Assume for a contradiction that [G0 , K] is trivial. Then we have [P, K, K] = 1, whence [P, K] = 1 according to Lemma 2.1.1. This contradicts the fact that CQ (P ) = 1; we conclude that CG0 (K) = 1 and [G0 , K] = G0 . Thus Q acts regularly on G0 . ¯ acts regularly on H 0 . We cannot apply the Now let us prove that Q same argument as for G, because H 0 is not abelian. But since (|P |, |Q|) = 1, asserting that Q acts regularly on G0 is equivalent to saying that all nontrivial conjugacy classes of G contained in G0 have length a multiple of |Q|. From the fact that the bijection α sends each conjugacy class of G contained in G0 onto a conjugacy class of H contained in H 0 , we deduce that all nontrivial ¯ conjugacy classes of H contained in H 0 have length a multiple of |Q| = |Q|. 0 ¯ This fact in turn is equivalent to saying that Q acts regularly on H . Thus ¯ H 0 (Q), ¯ assertion (iii) is proved. Furthermore, we have that H 0 = [H 0 , Q]C 0 ¯ ¯ according to Lemma 2.1.1 again. Since CH 0 (Q) = 1, it follows that [H , Q] = H 0. ¯ We have P = [P, Q]CP (Q). ¯ = CP¯ (Q). Now let us put W = CP (Q) and W Since G0 = [G0 , Q] ≤ [P, Q] ≤ G0 , we have [P, Q] = G0 , and thus P = G0 W . From G0 ∩ W = G0 ∩ CP (Q) = CG0 (Q) = 1 it follows that W is a complement for G0 in P . Now W is centralized by Q; consequently, W Q = W × Q is a ¯ is a complement for H 0 complement for G0 in G. We obtain similarly that W ¯Q ¯=W ¯ ×Q ¯ is a complement for H 0 in H. in P¯ and thus W ¯ are cyclic groups. The minimal normal Now let us show that Q and Q subgroup N of G, like any chief factor of G, can be regarded as an irreducible Fp G-module. By restriction N can be also regarded as an Fp Q-module, and N is still irreducible as an Fp Q-module. In fact any Fp Q-submodule of N is an Fp G-submodule, because G = P Q and N ≤ Z(P ). Furthermore, N is a faithful Fp Q-module, because Q acts regularly on G0 , and in particular on N . According to Theorem 2.3.1 then Q is cyclic of order dividing |N | − 1.

40


¯ is cyclic (this also follows directly from Similarly, we can deduce that Q ¯ ¯ acts the fact that Q and Q are both isomorphic to G/P ∼ = H/P¯ ), and that Q faithfully and irreducibly on H 00 . Thus assertions (ii) and (iv) are proved. It remains to prove assertion (v). If the element w of W centralized G0 , then w would be central in G, because W Q is an abelian complement for G0 in G. Hence hwi would be a normal subgroup of G not containing the unique minimal normal subgroup N . Therefore w = 1. Thus we have CW (G0 ) = 1. Similarly one proves that CW¯ (H 0 ) = 1. The proof of the lemma is now complete. ¤ ¯ ×Q ¯∼ Let us notice that W ×Q ∼ = G/G0 and W = H/H 0 . From the fact that G and H have identical character tables we deduce that G/G0 is isomorphic ¯ and Q ∼ ¯ to H/H 0 . It follows that W ∼ =W = Q. We also observe, as we already said in the proof of Lemma 3.2.1, that Q ¯ are non-trivial, otherwise G and H would be p-groups, contrary to and Q ¯ are not trivial either; if they were, Hypotheses 3.1.2. The groups W and W then G would have a normal abelian Sylow p-subgroup (namely G0 ) and H would not (because H 0 is not abelian). This would contradict the fact that G and H have the same character degrees; in fact [13, Corollary 12.34] asserts that a group has a normal abelian Sylow p-subgroup if and only if all its irreducible characters have degree not divisible by p.

3.3

Chief factors

We continue our analysis of groups G and H satisfying Hypotheses 3.1.2. In the next three lemmas we shall be concerned with certain chief factors of G and H regarded as irreducible G-groups and respectively H-groups by conjugation. Let M1 /M2 be a chief factor of G; hence M1 and M2 are normal subgroups of G with M2 ≤ M1 and there exists no normal subgroup M of G with M2 < M < M1 . Then M1 /M2 becomes an irreducible G-group by conjugation. Since the Fitting subgroup F(G) of G centralizes all chief factors of G (see [10, Kapitel III, Satz 4.3]), in particular F(G) is contained in the kernel of the action of G on M1 /M2 . Since Q is a complement for F(G) in G, the G-group M1 /M2 remains irreducible when it is regarded as a Q-group by restriction. Thus all chief factors of G are Q-composition factors of G (but not vice versa,

3.3. CHIEF FACTORS

41

because if M1 /M2 is a Q-composition factor of G the Q-subgroups M1 , M2 of G are not necessarily normal in G). Furthermore, two chief factors of G are G-isomorphic exactly when they are Q-isomorphic. Similar assertions hold ¯ for chief factors of H regarded as H-groups and as Q-groups. Lemma 3.3.1 Assume Hypotheses 3.1.2. Then all chief factors of G below G0 are G-isomorphic. Proof Let us regard the abelian normal subgroup G0 of G as a Q-group (actually a ZQ-module) by conjugation. In view of the remark which precedes the lemma it suffices to show that all composition factors of G0 are isomorphic as Q-groups. As we said in Section 2.4, there exists a unique decomposition of G0 (written additively) into the direct sum of its Q-homogeneous components B1 , . . . , Bm . Now W centralizes Q; consequently, each element of W induces by conjugation an automorphism of G0 as a Q-group. According to what we said in Section 2.4, it follows that each Bi is normalized by W , and being of course normal in G0 because G0 is abelian, each Bi is a normal subgroup of G. Since each non-trivial Bi contains the unique minimal normal subgroup N of G, there is a unique non-trivial Bi ; consequently, G0 is Q-homogeneous, that is to say, all Q-composition factors of G0 are Q-isomorphic. ¤ We shall also prove that when Hypotheses 3.1.2 hold, all chief factors of H below H 0 are H-isomorphic. We cannot use the same arguments as for G because H 0 is not abelian. Hence we shall split the proof in two parts: first we shall show that all chief factors of H lying between H 0 and H 00 are H-isomorphic and then that they are H-isomorphic to H 00 . In order to prove the first part we shall employ the fact that G and H have identical character tables, and hence in particular isomorphic lattices of normal subgroups. Lemma 3.3.2 Assume Hypotheses 3.1.2. Then all chief factors of H between H 0 and H 00 are H-isomorphic. Proof The statement of the lemma is clearly equivalent to the following assertion: all chief factors of H/H 00 below H 0 /H 00 are H-isomorphic. Since

42


¯ is a complement in H for the Fitting subgroup of H, which centralizes all Q ¯ chief factors of H, the proof will be complete once we show that the Q-group ¯ H 0 /H 00 is Q-homogeneous. We shall infer this from the fact that G0 /N is Qhomogeneous, by using the fact that G/N and H/H 00 have identical character tables. In fact, the character table isomorphism (α, β) from G to H induces ¯ from G/N to H/H 00 in a natural way. a character table isomorphism (¯ α, β) Let S be the product of all minimal normal subgroups of G/N contained in G0 /N (in other words, S is the socle of G0 /N as a ZG-module). Since the map α ¯ induces an isomorphism between the lattices of normal subgroups of G/N and H/H 00 , and sends G0 /N onto H 0 /H 00 , the image S¯ of S under α is the product of all minimal normal subgroups of H/H 00 contained in H 0 /H 00 . The groups S and S¯ are elementary abelian p-groups, and thus can be regarded as an Fp G-module and an Fp H-module respectively. The Fp Gsubmodules of S (that is to say, the normal subgroups of G/N contained in S) are in a bijective correspondence, induced by the map α ¯ , with the Fp H¯ submodules of S. We also observe that the Fitting subgroups of G and H centralize S and S¯ respectively; it follows that the set of Fp G-submodules of S coincides with the set of Fp Q-submodules of S. Of course, a similar assertion holds for H. As a consequence, the map α ¯ induces a bijection from ¯ ¯ the set of Fp Q-submodules of S onto the set of Fp Q-submodules of S. We know that the semisimple Fp Q-module S is homogeneous; according to Lemma 2.4.1, this is equivalent to the fact that S is the (set-theoretic) union of its irreducible Fp Q-submodules. This property of S can clearly be passed on to S¯ via the bijection α ¯ , namely we get that S¯ is the union of ¯ its irreducible Fp Q-submodules. Then Lemma 2.4.1 again yields that S¯ is ¯ Fp Q-homogeneous. ¯ From this fact we deduce that H 0 /H 00 is Q-homogeneous, as follows. The 0 00 ¯ abelian Q-group H /H can be decomposed into the direct product of its ¯ ¯ Q-homogeneous components (see Section 2.4). If M is a non-trivial Q0 00 ¯ homogeneous component of H /H , then M is normalized by W . In fact, ¯ commutes with the elements of Q, ¯ and therefore induces every element of W 0 00 ¯ a Q-automorphism of H /H by conjugation; on the other hand, M is left in¯ ¯ . It variant by every Q-endomorphism of H 0 /H 00 , and thus is normalized by W follows that M is a non-trivial normal subgroup of H 0 /H 00 . In particular, M ¯ intersects S¯ non-trivially. Because we have proved that S¯ is Q-homogeneous, 0 00 ¯ it follows that H /H has only one Q-homogeneous component. In other ¯ words, H 0 /H 00 is Q-homogeneous, and the proof is complete. ¤

3.3. CHIEF FACTORS

43

It remains to show that some chief factor of H between H 0 and H 00 is H-isomorphic to H 00 . As we have already remarked in several places, since ¯ complements the Fitting subgroup P¯ of H in H, it is sufficient to find a Q ¯ Q-isomorphism from a chief factor of H between H 0 and H 00 onto H 00 . The tool which will produce such an isomorphism is Lemma 2.2.4. ¯ The map from H 0 to itself which ¯ = CP¯ (Q). Let us fix an element w ¯ of W sends x to [x, w] ¯ is not a group homomorphism, and thus its image bH 0 , wc ¯ is not necessarily a subgroup of H 0 . However, according to Lemma 2.2.4 and the discussion which precedes it, the inverse image of H 00 under this map, namely ¯ w¯ = {x ∈ H 0 | [x, w] M ¯ ∈ H 00 }, is a subgroup of H 0 , and the restriction ¯ w¯ → H 00 ϕ¯w¯ : M x 7→ [x, w] ¯ ¯ Q-homomorphism ¯ of our map is a W with kernel CH 0 (w). ¯ From the fact that 00 ¯ w¯ and CH 0 (w) H ≤ Z(P¯ ) it follows that the subgroups M ¯ of H 0 contain H 00 ; therefore, they are normal in H 0 , because H 0 /H 00 is abelian. The fact that ¯Q ¯ finally implies that they are normal subgroups they are normalized by W of H. ¯ Q-homomorphism ¯ Let us suppose for a moment that our W ϕ¯w¯ is surjec¯ Q-isomorphism ¯ ¯ w¯ /CH 0 (w), tive. Then ϕ¯w¯ induces a W from M ¯ which therefore is a chief factor of H, onto H 00 . Thus the assertion that some chief factor ¯ Q-isomorphic ¯ of H between H 0 and H 00 is W to H 00 is proved, provided we can show that H 00 ⊆ bH 0 , wc, ¯ or equivalently that H 00 ⊆ bw, ¯ H 0 c, for some ¯. w¯ ∈ W We shall infer this fact from a corresponding fact for G by means of the character table isomorphism (α, β). Indeed, we shall prove the following lemma. Lemma 3.3.3 Assume Hypotheses 3.1.2. Then wN ⊆ wG for all w ∈ W \1, ¯ \ 1. and wH ¯ 00 ⊆ w¯ H for all w¯ ∈ W Proof Let us fix w ∈ W = CG0 (Q) and let us consider the map ϕw : G0 → G0

44


defined by xϕw = [x, w]. According to Lemma 2.2.3 (with H1 , H2 , K1 , K2 , w, Q replaced by G0 , G0 , 1, 1, w, W Q respectively), the map ϕw is a W Qhomomorphism. Thus the situation here is much better than it was for H. In fact, the image (G0 )ϕw = bG0 , wc of ϕw is a W Q-subgroup of G0 ; hence it is a normal subgroup of G, because G0 is abelian and G = G0 W Q. Now let us assume that w 6= 1. Then w does not centralize G0 , because we know from Lemma 3.2.1 that W acts faithfully on G0 by conjugation. Hence ϕw is not the trivial homomorphism. In particular, its image bG0 , wc = [G0 , w] is a non-trivial normal subgroup of G, and therefore it contains the unique minimal normal subgroup N . This is clearly equivalent to N ⊆ bw, G0 c, 0 and eventually to wN ⊆ wG . This clearly implies the first assertion of the lemma. In order to pass on this piece of information to H we shall first express it in character-theoretical language. According to Lemma 4.2.1, the statement wN ⊆ wG for all w ∈ W \ 1 is equivalent to χ(w) = 0 for all χ ∈ Irr(G) \ Irr(G/N ) and for all w ∈ W \ 1. We must be cautious here in applying the character table isomorphism (α, β), ¯ , indeed, W α is not necbecause W α does not necessarily coincide with W essarily a subgroup of H. But we observe that since characters are class functions, the statement χ(g) = 0 for all χ ∈ Irr(G) \ Irr(G/N ) actually holds for all g ∈ G which are conjugate to some w ∈ W \ 1. Now we claim that the set of elements g ∈ G which are conjugate to some element of W coincides with the set of elements g ∈ P such that |Q| divides |CG (g)| (and a similar assertion holds for H). In fact, if g ∈ G is conjugate to w ∈ W , then |CG (g)| = |CG (w)|, and |Q| divides |CG (w)| because Q ≤ CG (w). On the other hand, suppose that |Q| divides the order of the centralizer in G of an element g of G. According to the Theorem of Schur-Zassenhaus, there exists a complement Q0 for P ∩ CG (g) in CG (g). Since |G|p0 = |Q| divides |CG (g)|, we have |Q0 | = |Q|; therefore Q0 is a complement for P in G. The conjugacy part of the Theorem of Schur-Zassenhaus

3.3. CHIEF FACTORS

45

yields now that Q = Qx0 for some x ∈ G. Since obviously g ∈ CP (Q0 ), it follows that g x ∈ CP (Qx0 ) = CP (Q) = W , and therefore g is conjugate to some element of W . Our claim is proved. The analogous claim for H can be proved similarly. Thus we have χ(g) = 0 for all χ ∈ Irr(G) \ Irr(G/N ), for all g ∈ P \1 such that |Q| divides |CG (g)|. Let us apply the character table automorphism (α, β) to this statement. After noticing that P α = P¯ , and that |CH (g α )| = |CG (g)| for all g ∈ G (according to the second orthogonality relation), we obtain χ(h) = 0 for all χ ∈ Irr(H) \ Irr(H/H 00 ), ¯ divides |CH (h)|. Since in particular |Q| ¯ divides for all h ∈ P¯ \ 1 such that |Q| ¯ , an application of Lemma 4.2.1 yields |CH (w)| ¯ for all w ¯∈W ¯ \ 1, wH ¯ 00 ⊆ w¯ H for all w¯ ∈ W which concludes the proof.

¤

Now the second assertion of Lemma 3.3.3 together with the fact that ¯ 6= 1 (see the note after Lemma 3.2.1) implies that there exists some W ¯ \ 1 such that H 00 ⊆ bH, wc. ¯ QH ¯ 0 and [W ¯ Q, ¯ w] w¯ ∈ W ¯ Since H = W ¯ = 1, we 0 ¯ ¯ have bH, wc ¯ = bH , wc. ¯ It follows that the W Q-homomorphism ¯ w¯ → H 00 ϕ¯w¯ : M x 7→ [x, w], ¯ which we defined in the discussion preceding Lemma 3.3.3, is an epimorphism, and thus induces an H-isomorphism of some chief factor of H between H 0 and H 00 onto H 00 . Thus we have the following lemma. Lemma 3.3.4 Assume Hypotheses 3.1.2. Then all chief factors of H below H 0 are H-isomorphic. Proof The assertion follows from the preceding discussion together with Lemma 3.3.2. ¤

46


Let us remark again that since Q is a complement for the Fitting subgroup F(G) in G, any chief factor of G is irreducible as a Q-group and is therefore a composition factor of G as a Q-group (similarly for H). Hence Lemma 3.3.1 and Lemma 3.3.4 imply that all Q-composition factors of G0 are Q-isomorphic ¯ ¯ and respectively that all Q-composition factors of H 0 are Q-isomorphic.

3.4

¯ on H 0 More about the action of Q

¯ In the previous section we have proved in particular that all Q-composition 0 ¯ ¯ on factors of H are Q-isomorphic. On the other hand, the actions of Q ¯ on H 0 /Z(H 0 ) H 0 /H 00 and on H 00 are not independent: in fact the action of Q 00 ¯ on H via the operation of forming determines uniquely the action of Q commutators. In this section we shall play these two facts against each other ¯ on H 0 . This in order to draw further consequences about the action of Q will require the machinery developed in Sections 2.2 and 2.3, but the simpler reasoning of the next proposition will give the reader a taste of the method. First let us recall a basic fact about automorphisms of cyclic p-groups: if C is a cyclic group of order pe (p a prime), and ψ¯ is an automorphism of p0 -order of C/Φ(C), then there is a unique automorphism ψ of p0 -order of C ¯ In fact, ψ¯ has the form which induces ψ. ¯

cψ = ca mod Φ(C) for all c ∈ C, for some integer a (which is unique if we require 0 < a < p). The map ψˆ : C → C defined by ˆ cψ = ca for all c ∈ C, is an automorphism of C, because p does not divide a. However, ψˆ might have order divisible by p. According to [10, Kapitel I, Satz 4.6 and Satz 13.19], the group of automorphisms of C is abelian (cyclic if p 6= 2) of order (p − 1)pe−1 , hence if f is any integer greater than or equal to e − 1, then the f automorphism ψ = ψˆp of C has p0 -order. We clearly have cψ = cb for all c ∈ C, f

where b = ap . Since b ≡ a (mod p), the automorphism of C/Φ(C) induced ¯ The uniqueness of ψ follows from the fact that C and C/Φ(C) by ψ is ψ. have the same number of automorphisms of p0 -order, namely p − 1.

¯ ON H 0 3.4. MORE ABOUT THE ACTION OF Q

47

Proposition 3.4.1 Assume Hypotheses 3.1.2. Then H 00 is not cyclic. Proof Let us assume for a contradiction that H 00 is cyclic, whence it has ¯ and let a be the unique integer order p. Let us choose a generator η of Q with 1 ≤ a < p such that z η = z a for all z ∈ H 00 . The abelian factor group H 0 /H 00 has a decomposition into a direct product ¯ of Q-indecomposable groups. Let C be one of them. According to Theorem ¯ 2.4.2, the group C is homocyclic and C/Φ(C) is an irreducible Q-group; 0 00 ¯ ¯ hence C/Φ(C) is a Q-composition factor of H /H . Since all Q-composition ¯ factors of H 0 /H 00 are Q-isomorphic to H 00 by Lemma 3.3.4, we have that C/Φ(C) is cyclic of order p. Consequently, C is cyclic. Because C/Φ(C) is ¯ Q-isomorphic to H 00 , we have cη ≡ ca mod Φ(C) for all c ∈ C. If pe is the exponent of H 0 /H 00 , then |C| divides pe . Since η acts on C as an automorphism of order prime to p, the remark which precedes this proposition yields cη = cb for all c ∈ C, e−1 ¯ where b = ap . Now this holds for all Q-indecomposable groups C of which 0 00 H /H is the direct product, and therefore we have xη = xb mod H 00 for all x ∈ H 0 . We also have z η = z b for all z ∈ H 00 , because b ≡ a (mod p). Now let us choose x, y ∈ H 0 such that [x, y] 6= 1, whence h[x, y]i = H 00 . Remembering that H 00 ≤ Z(H 0 ), we compute 2

[x, y]η = [xη , y η ] = [xb , y b ] = [x, y]b . On the other hand, since [x, y] ∈ H 00 we have [x, y]η = [x, y]b .

48


Since [x, y] has order p, it follows that b2 ≡ b (mod p); it follows that b ≡ 1 (mod p), because b is not divisible by p. But this implies that z η = z b = z for all z ∈ H 00 , ¯ or in other words, that η centralizes H 00 . This contradicts the fact that Q 00 acts faithfully on H , by Lemma 3.2.1. Hence our assumption is wrong, and thus H 00 cannot be cyclic. ¤ The key idea of the proof of Proposition 3.4.1, namely that the action of ¯ Q on H 0 must be compatible with commutation, will now be generalized to the case in which H 00 is not assumed to be cyclic. However, this will not lead to any contradiction in general. Let us assume Hypotheses 3.1.2, and let us fix a chief series of H going from 1 to H 0 thus: 1 < H 00 = K1 < K2 < · · · < Kt = H 0 . Let r be the smallest index i such that Ki 6≤ Z(H 0 ) (since H 0 is not abelian such an index exists), and then let s be the smallest index j such that [Kr , Kj ] 6= 1. Then we clearly have 1 < r ≤ s ≤ t. Since [Kr , Ks ] ≤ H 00 ≤ Z(H 0 ) and [Kr , Ks−1 ] = [Kr−1 , Ks ] = 1, according to Lemma 2.2.1 the map γ : Kr /Kr−1 × Ks /Ks−1 → H 00 such that (xKr−1 , yKs−1 )γ = [x, y] for all x ∈ Kr and for all y ∈ Ks , is Z-bilinear. Moreover, if r = s, the map γ is skew-symmetric. Since Kr /Kr−1 , Ks /Ks−1 and H 00 have exponent p, they can be regarded as vector spaces over Fp , and the map γ is Fp -bilinear. We shall put V1 = Kr /Kr−1 , V2 = Ks /Ks−1 and V = H 00 . Let V1 ⊗ V2 denote the tensor product of V1 and V2 over Fp . According to the universal property of tensor products, there exists a unique Fp -linear map γ¯ : V1 ⊗ V2 → V such that (v1 ⊗ v2 )γ¯ = (v1 , v2 )γ for all v1 ∈ V1 and for all v2 ∈ V2 .

¯ ON H 0 3.4. MORE ABOUT THE ACTION OF Q

49

¯ Now V1 , V2 and V are irreducible Fp Q-modules by conjugation, and according to Lemma 3.3.4, they are all isomorphic. ¯ The tensor product V1 ⊗V2 becomes an Fp Q-module in a standard way (see for instance [13, Chapter 4], [10, Kapitel V, Definition 10.4], or [5, Definition ¯ on V1 ⊗ V2 is defined on (10.15)]). In short, the action of a generator η of Q the pure tensors v1 ⊗ v2 (for vi ∈ Vi ) by the formula (v1 ⊗ v2 )η = v1 η ⊗ v2 η and extended Fp -linearly to V1 ⊗ V2 . (Although the module action in this case is given by conjugation, instead of the exponential notation v1η we shall use the more common notation v1 η.) Since [xη , y η ] = [x, y]η for all x ∈ Kr and y ∈ Ks , we have ((v1 ⊗ v2 )η)γ¯ = ((v1 ⊗ v2 )γ¯ )η for all pure tensors v1 ⊗ v2 ; hence it follows by linearity that (wη)γ¯ = (wγ¯ )η for all w ∈ V1 ⊗ V2 . ¯ Thus γ¯ : V1 ⊗ V2 → V is an Fp Q-module homomorphism. It cannot be the zero homomorphism, because [Kr , Ks ] 6= 1. But V is an irreducible ¯ Fp Q-module, and therefore γ¯ is an epimorphism. Hence the tensor square ¯ Fp Q-module V ⊗ V (which is isomorphic to V1 ⊗ V2 ) has some factor module isomorphic to V . This is impossible when V has dimension 1 over Fp (unless V is the trivial module, which it is not in our case, according to Lemma 3.2.1). In fact, if V has dimension 1, then V ⊗ V also has dimension 1; in particular V ⊗ V is irreducible, but it cannot be isomorphic to V , unless V is the trivial module. For, let v be a generator of V ; hence v ⊗ v generates V ⊗ V . We have vη = av for some a ∈ F× p (which is clearly independent of the choice of v), and thus (v ⊗ v)η = vη ⊗ vη = av ⊗ av = a2 (v ⊗ v). It is clear then that V and V ⊗ V are not isomorphic unless a = a2 , that is to say a = 1 (because a 6= 0), which means that V is the trivial module. Thus we obtain a different formulation of the proof of Proposition 3.4.1. In the next section we shall examine when V ⊗ V has a factor module isomorphic to V , for small values of the dimension of V over Fp . But before

50


doing that, we observe that in the special case in which r = s we have V1 = V2 , and the bilinear map γ is skew-symmetric, which means (v, v)γ = 0 for all v ∈ V1 . The subspace W = hv ⊗ v ∈ V1 ⊗ V1 | v ∈ V1 i ¯ of V1 ⊗V1 , which is clearly an Fp Q-submodule of V1 ⊗V1 , is therefore contained ¯ in the kernel of the Fp Q-module epimorphism γ¯ . The factor space V1 ⊗ V1 /W is by definition the exterior square V1 ∧ V1 , ¯ which thus becomes an Fp Q-module (with the notation of [11, Chapter VII, Definition 8.16] and according to [11, Chapter VII, Lemma 8.17], we have V1 ∧ V1 = V1 ⊗ V1 /S(V1 ) ∼ = A(V1 ); see also [5, §12A]). ¯ Hence we obtain an Fp Q-module epimorphism γ¯ : V1 ∧ V1 → V , such that ¯

(v1 ∧ v2 )γ¯ = (v1 , v2 )γ for all v1 , v2 ∈ V1 , where by definition v1 ∧ v2 = (v1 ⊗ v2 ) + W ∈ V1 ∧ V1 . ¯ Therefore, when r = s, the Fp Q-module V is isomorphic to a factor module of V ∧ V . We shall henceforth distinguish between the case in which r < s (which only gives rise to a map γ¯ : V1 ⊗ V2 → V ) and the case in which r = s (which in addition gives rise to a map γ¯ : V1 ∧ V1 → V ) by referring to them as the binary case and the unary case.

3.5

The smallest cases which can occur

¯ the modules In this section we shall investigate for which values of p and |Q| V ⊗ V and V ∧ V have a composition factor isomorphic to V , where V is a ¯ over Fp . faithful irreducible module for the cyclic p0 -group Q Let E be a splitting field for the polynomial x|Q| − 1 over Fp ; hence E ¯ is the smallest extension field of Fp which contains a primitive |Q|th root of unity, or, in other words, the smallest extension field of Fp whose mul¯ (which is necessarily tiplicative group E× contains a subgroup of order |Q|

3.5. THE SMALLEST CASES WHICH CAN OCCUR

51

cyclic). Hence E is clearly isomorphic to Fpn , where n is the multiplicative ¯ that is to say, n is the smallest positive integer such order of p (mod |Q|), ¯ divides pn − 1. According to Theorem 2.3.1, the integer n equals that |Q| the dimension of V over Fp . ¯ is not divisible by p, the Fp -algebra Fp Q ¯ is semisimple, according Since |Q| ¯ to Maschke’s Theorem. In particular, V ⊗ V and V ∧ V are semisimple Fp Qmodules. According to Corollary 2.3.3, the composition factors of V ⊗ V and ¯ V ∧ V as Fp Q-modules are completely determined by the composition factors E ¯ of (V ⊗ V ) and (V ∧ V )E as EQ-modules. It is easy to see that (V ⊗ V )E ∼ = VE⊗VE

and (V ∧ V )E ∼ = V E ∧ V E.

¯ Let ε be a primitive |Q|th root of unity in E. The discussion which follows n−1 Theorem 2.3.1 yields that ε, εp , . . . , εp are all the distinct eigenvalues of η E E ¯ (a generator of Q) on V . Moreover, V has a basis v0 , . . . , vn−1 over E such that i vi η = εp vi , for all i = 0, . . . , n − 1. Thus bases for V E ⊗ V E and V E ∧ V E over E are given by vi ⊗ vj , for i, j = 0, . . . , n − 1, and respectively vi ∧ vj , for 0 ≤ i < j < n. These bases are made of eigenvectors for η; in fact i

j

(vi ⊗ vj )η = vi η ⊗ vj η = εp +p (vi ⊗ vj ) and

i

j

(vi ∧ vj )η = vi η ∧ vj η = εp +p (vi ∧ vj ). The eigenvalues for η on V E ⊗V E and V E ∧V E (considered with multiplicities) can be grouped into Galois conjugacy classes, which in turn determine the isomorphism classes of the composition factors of V E ⊗ V E and respectively ¯ according to Theorem 2.3.2. of V E ∧ V E as Fp Q-modules, Sets of representatives of the distinct Galois conjugacy classes of the eigenvalues of η on V E ⊗ V E and V E ∧ V E are contained in the set i

{εp +1 | 0 ≤ i ≤ n/2}

52


and respectively in the set i

{εp +1 | 0 < i ≤ n/2}. i

j

j−i

i

j

n−i

In fact, the eigenvalue εp +p is Galois conjugate to εp +1 = (εp +p )p and n+i−j +1 i j n−j to εp = (εp +p )p ; on the other hand, if 0 ≤ i ≤ j < n, then we have either 0 ≤ j − i ≤ n/2 or 0 ≤ n + i − j ≤ n/2. Hence V ⊗ V has a composition factor (hence a direct summand, because j V ⊗V is semisimple) isomorphic to V exactly when εp appears as an element i of the set {εp +1 | 0 ≤ i ≤ n/2} for some j = 0, . . . , n − 1. A similar assertion holds for V ∧ V . We shall state both assertions in the following lemma. ¯ Lemma 3.5.1 Let V be a faithful irreducible module for a cyclic group Q ¯ over Fp (in particular p does not divide |Q|) and let n be the dimension of V over Fp . Then: ¯ (i) the tensor square Fp Q-module V ⊗ V has a direct summand isomorphic to V if and only if ¯ pi + 1 ≡ pj mod |Q| for some i, j with 0 ≤ i ≤ n/2 and 0 ≤ j < n (or, equivalently, for some non-negative integers i, j); ¯ (ii) the exterior square Fp Q-module V ∧V has a direct summand isomorphic to V if and only if ¯ pi + 1 ≡ pj mod |Q| for some i, j with 0 < i ≤ n/2 and 0 < j < n (or, equivalently, for some non-negative integers i, j, with i not a multiple of n). ¯ odd in We observe here that we may also restrict our attention to |Q| ¯ Lemma 3.5.1. In fact, if |Q| is even, then p is odd, and hence the congru¯ has clearly no solution. It follows in particular ence pi + 1 ≡ pj (mod |Q|), that if (G, H) is a pair of groups which satisfy Hypotheses 3.1.2, then the ¯ for the normal Sylow p-subgroup of H has odd order. cyclic complement Q However, this is also a consequence of the general fact that a non-abelian group cannot have a fixed-point-free automorphism of order 2, according to ¯ induces a [10, Kapitel V, Satz 8.18] (while any non-identity element of Q 0 fixed-point-free automorphism of the non-abelian group H by conjugation).


53

We shall see in Chapter 5 that all the cases described in Lemma 3.5.1 actually arise from the analysis of counterexamples to Conjecture 3.1.1. In other words, for any faithful irreducible module V for a non-trivial cyclic ¯ over Fp such that V ⊗V (respectively V ∧V ) has a composition factor group Q isomorphic to V , there exists a pair of groups (G, H) satisfying Hypotheses 3.1.2 and such that ¯ is a complement for the normal Sylow p-subgroup of H, • Q ¯ • H 00 is isomorphic to V when it is regarded as an Fp Q-module by conjugation, • the map γ¯ (respectively γ¯ ) arising from some chief series of H, as ¯ described in the last section, is an Fp Q-module epimorphism of V ⊗ V (respectively of V ∧ V ) onto V . Now let us take a different point of view. We shall explicitly construct pairs (G, H) of groups satisfying Hypotheses 3.1.2 in Chapter 5: we would like them to be as small as possible. Therefore it makes sense to fix a small value n of the dimension of V and then determine for which primes p and ¯ some (actually, according to Lemma 3.5.1, any) faithful cyclic p0 -groups Q ¯ over Fp appears as a composition factor of V ⊗V , irreducible module V for Q or even of V ∧ V . We shall see in some detail what happens for n = 1, 2, 3, 4 and show in particular that while case (i) of Lemma 3.5.1 can already happen for n = 2, case (ii) does not occur unless n ≥ 4. Case n = 1. We have already seen that in this case V ⊗ V (which is irreducible, because it has dimension 1 over Fp ) cannot be isomorphic to V ¯ as an Fp Q-module. Case n = 2. Since in this case V ∧ V has dimension 1, it is irreducible and certainly not isomorphic to V , which has dimension 2. On the other hand, V ⊗ V has dimension 4 over Fp . According to Lemma 3.5.1 then V ⊗ V has a composition factor isomorphic to V if and only if ¯ for some i = 0, 1 and j = 0, 1. The cases (i, j) = pi + 1 ≡ pj (mod |Q|) (0, 0), (1, 0), (1, 1) are easily ruled out, remembering that p does not divide ¯ Hence we are left with p0 + 1 = p (mod |Q|). ¯ |Q|. If this is the case, from ¯ ¯ |Q| | (p − 2) it follows that (|Q|, p − 1) = 1. Now Fp2 contains a primitive ¯ ¯ divides |Q|th root of unity according to Theorem 2.3.1; consequently, |Q|

54


2 ¯ |F× p2 | = p − 1 = (p − 1)(p + 1). Therefore |Q| divides both p + 1 and p − 2, ¯ = 3. and hence |Q| Thus there exists a two-dimensional faithful irreducible module V for a ¯ over Fp such that V is isomorphic to a composition factor of cyclic group Q ¯ = 3 and p ≡ −1 (mod 3). V ⊗ V , if and only if |Q|

Case n = 3. It is certainly possible that V is isomorphic to a composition ¯ = 7 and p ≡ 4 (mod 7) we have factor of V ⊗ V . For instance, when |Q| ¯ and that p0 + 1 ≡ p2 (mod |Q|). ¯ that p has multiplicative order 3 (mod |Q|) However, we shall not go into further details here. We shall only prove that V ∧ V cannot have any composition factor isomorphic to V . In fact, if this ¯ were true we would have, according to Lemma 3.5.1, that p + 1 ≡ pj mod |Q| for some j = 0, 1, 2. Since the cases j = 0, 1 are easily ruled out, we would ¯ divides p2 −p−1. On the other hand, because Fp3 must contain have that |Q| ¯ ¯ should divide p3 − 1. It would follow that a primitive |Q|th root of unity, |Q| 3 2 ¯ |Q| divides (p − 1) − (p + 1)(p − p − 1) = 2p, and this contradicts the fact ¯ is odd and prime to p. that |Q| Case n = 4. We shall prove that if V has dimension 4 it can happen that V is a composition factor of V ∧ V (and hence of V ⊗ V too). According to ¯ it is possible Lemma 3.5.1 we need to determine for which values of p and |Q| i j ¯ to have p + 1 ≡ p (mod |Q|) for some i = 1, 2 and j = 0, 1, 2, 3. It is not difficult to rule out the cases with i = 2, either by working with 2 3 the congruences or by simply noticing that the eigenvalues εp +1 and εp +p of ¯ on V ∧ V form a Galois conjugacy class of length two over a generator η of Q Fp , which thus corresponds to a composition factor of V ∧ V of dimension two, in particular not isomorphic to V , which has dimension n = 4. Hence we are left with i = 1. Now the cases (i, j) = (1, 0), (1, 1) are clearly impossible; hence we have either (i, j) = (1, 2) or (i, j) = (1, 3). In the first case, we have that p+1 ≡ p2 ¯ in other words |Q| ¯ divides p2 − p − 1. Keeping in mind that |Q| ¯ (mod |Q|), × 4 ¯ also divides p − 1 = |Fp4 |, we obtain that |Q| divides (p4 − 1) − (p2 − p − 1)(p2 + 1) = p(p2 + 1) ¯ divides p2 + 1. Hence |Q| ¯ also divides and therefore that |Q| (p2 + 1) − (p2 − p − 1) = p + 2.


55

¯ and p ≡ −2 (mod |Q|) ¯ we obtain that 5 ≡ 0 Now from p2 ≡ −1 (mod |Q|) ¯ ¯ ¯ (mod |Q|). It follows that |Q| = 5 and p ≡ 3 (mod |Q|). In a similar way ¯ one can find that in the case (i, j) = (1, 3) it must be |Q| = 5 and p ≡ 2 ¯ (mod |Q|). ¯ = 5 and p ≡ 3 (mod |Q|) ¯ or p ≡ 2 (mod |Q|), ¯ then p Conversely, if |Q| 2 ¯ and p + 1 ≡ p (mod |Q|), ¯ or respechas multiplicative order 4 (mod |Q|) 3 ¯ tively p + 1 ≡ p (mod |Q|). We conclude that there exists a faithful irreducible module V of dimension ¯ such that V is isomorphic to a composition 4 over Fp for a cyclic group Q, ¯ = 5 and p ≡ 2 or 3 (mod 5). factor of V ∧ V , exactly when |Q|

Chapter 4 Comparing character tables 4.1

Our philosophy

We said in Chapter 1 that two groups can have identical character tables without being isomorphic. But how can one compare character tables in practice? In this chapter we shall develop a method which allows one to do this, in special situations. We shall employ some basic Clifford theory. This is the part of character theory which analyzes the relations between the characters of a group G and the characters of a normal subgroup N of G. For our present purposes, the two most fundamental results of Clifford theory will suffice, namely Clifford’s Theorem (see [13, Theorem 6.2]), and the Clifford Correspondence ([13, Theorem 6.11]). Let us start from the side of group theory by recalling the well-known analysis of a group G in terms of a normal subgroup N of G and the factor group G/N . Given two groups N and H, there are in general many ways of constructing a group G which has N as a normal subgroup and such that G/N ∼ = H. This is the so-called extension problem for groups, and its answer is given by the following well-known theorem. Theorem 4.1.1 Let H and N be groups, let h 7→ ϕ(h) be a map from H into Aut(N ), and let (h1 , h2 ) 7→ f (h1 , h2 ) 56

4.1. OUR PHILOSOPHY

57

be a map from H × H into N , a so-called factor system. Let us assume that ϕ and f satisfy the following conditions, for all n ∈ N and for all hi , h ∈ H: (1) f (h1 , h2 h3 )f (h2 , h3 ) = f (h1 h2 , h3 )f (h1 , h2 )ϕ(h3 ) , (2) nϕ(h1 )ϕ(h2 ) = (nϕ(h1 h2 ) )f (h1 ,h2 ) , (3) f (h, 1) = f (1, h) = 1. Let us define a multiplication on the cartesian product G = {(h, n) | h ∈ H, n ∈ N }, through the formula ϕ(h2 )

(h1 , n1 )(h2 , n2 ) = (h1 h2 , f (h1 , h2 )n1

n2 ).

Then G becomes a group with this multiplication. The set ¯ = {(1, n) | n ∈ N } N ¯ is is a normal subgroup of G isomorphic to N , and the factor group G/N isomorphic to H. The group G is called the extension of H by N with respect to the automorphisms ϕ(h) and the factor set f ( , ). Proof See [10, Kapitel I, Satz 14.2], or [8, Theorem 15.1.1].

¤

Let us summarize Theorem 4.1.1 by saying that in order to construct the group G from the normal subgroup N and the factor group G/N , we need the following ingredients: (i) the group H = G/N , (ii) the group N , together with a map ϕ : H → Aut(N ), (iii) a factor set f : H × H → N ; moreover, conditions (1), (2), and (3) of Theorem 4.1.1 have to be satisfied. Let us say that our ingredient (iii) is usually the most difficult to handle, and can be dealt with by using cohomological methods.

58

CHAPTER 4. COMPARING CHARACTER TABLES

Let us remark that in the special case in which N is abelian, and more generally when f (h1 , h2 ) ∈ Z(N ) for all h1 , h2 ∈ H, the map h 7→ ϕ(h) is a group homomorphism from H into Aut(N ); thus when N is abelian, it becomes a ZH-module. When N is arbitrary, the map ϕ is not a group homomorphism; however, the composite map ϕπ : H → Out(N ) = Aut(N )/ Inn(N ) is a homomorphism, where π : Aut(N ) → Aut(N )/ Inn(N ) is the natural epimorphism. As a consequence, ϕ induces an action of H on the set of conjugacy classes of N . The knowledge of the orbits of this action allows one to determine the conjugacy classes of G which are contained in N . More information is needed in general in order to determine the remaining conjugacy classes of G, namely some information about the factor set f is necessary. Let us turn our attention to the character tables now. Let us order the conjugacy classes of G in such a way that those which are contained in the normal subgroup N precede those which are contained in G\N . Similarly, for the irreducible characters of G, let us first list those whose kernel contains N , which we shall identify with the characters of G/N , and then the remaining ones. The character table T of G can be divided into four submatrices accordingly, thus: "N Irr(G/N ) Irr(G) \ Irr(G/N )

G\N

A B C D

#

Let us examine which of the submatrices A, B, C, D of T is influenced by each of our ingredients (i), (ii), and (iii). Ingredient (i), namely the factor group H = G/N , gives information about A and B. In fact, the submatrix of T made up of the submatrices A and B coincides with the character table T¯ of the factor group G/N , except that some columns of T¯ are repeated in A or B (in particular all the columns of A are equal). Therefore (i) determines the number of rows of A and B, and all the entries, up to repeating some columns. Now let us see how the normal subgroup N together with the map ϕ : H → Aut(N ), which constitute our ingredient (ii), give information about

4.1. OUR PHILOSOPHY

59

A and C. The matrices A and C display the restrictions of the irreducible characters of G to the normal subgroup N . According to Clifford’s Theorem, if χ is an irreducible character of G, then its restriction χN is a multiple of the sum over a G-orbit of irreducible characters of N . We are assuming that the group N is known; consequently, its character table is uniquely determined. We have already said that the map ϕ determines an action of H = G/N on the set of conjugacy classes of N ; to be explicit, if h ∈ H and K is a conjugacy class of N , then Kh is the conjugacy class of N given by Kh = {nϕ(h) | n ∈ K}. The map ϕ also determines an action of H on Irr(N ) via −1

θh (n) = θ(nϕ(h) ) for all n ∈ N, where h ∈ H and θ ∈ Irr(N ). In other words, the map ϕ induces two actions of H, one on the set of columns and one on the set of rows of the character table of N . Therefore, assuming that we are able to write the character table of N , we can replace each set of rows of the table which correspond to a G-orbit of Irr(N ) with a single row, namely their sum; finally, we can delete multiple columns. Let us call T˜ the matrix which we obtain. Then, first of all, A and C have the same number of columns as T˜. Secondly, possibly after permuting the columns of T˜, each row of A or C is a multiple of some row of T˜ by a positive integer; on the other hand, each row of T˜ has some multiple which appears as a row of either A or C. In other words, the submatrix of T made up of A and C has the same number of columns as T˜, and can be obtained from T˜ by repetition of some rows and then multiplication of some rows by some positive integers. The little asymmetry in the ways in which we obtained A, B from T¯ and A, C from T˜ would disappear if we considered the table of central characters ωχ of G, instead of the ordinary character table of G; here the central character ωχ associated with the irreducible character χ of G is the map from G into C defined by the formula ωχ (g) =

χ(g) for all g ∈ G χ(1)

(the name central character is due to the fact that ωχ can be extended Clinearly to a character of the centre Z(CG) of the group algebra CG, in other words, to a C-algebra homomorphism from Z(CG) into C).

60


We have seen that the submatrix A of T is completely determined by our ingredients (i) and (ii), while B and C are only partly determined (B is determined up to repeating columns, and C up to repeating rows and multiplying them by positive integers). The remaining submatrix of T , namely D, usually requires some knowledge of our ingredient (iii). We shall get rid of the problem of computing D by assuming that D is the zero matrix. As we shall see, this assumption will also eliminate the residual indeterminacy in the matrices B and C; in fact there will not be any repeated column in B, nor any repeated row in C, and the first orthogonality relation will determine which multiple of each row of T˜ (not corresponding to the trivial character of N ) appears as a row of C. Thus our ingredients (i) and (ii), together with the assumption D = 0, will determine the character table of G uniquely (though they do not determine G up to isomorphism). We shall see in Section 4.2 how the condition D = 0 can be expressed by two equivalent statements: one of them concerns conjugacy classes of G and G/N , while the other one concerns irreducible characters of G and N . We observe now that in order to describe the submatrices A, B, and C of T we did not use all of the information contained in our ingredients (i) and (ii). In fact, in our discussion we never used the group structure of H and N , but only their character tables, together with the orbits of H on the set of conjugacy classes and the set of irreducible characters of N . It turns out that our ingredients (i) and (ii) can be safely replaced with the following weaker ingredients: (I) the character table of H = G/N , (II) the character table of N , together with the knowledge of the G-orbits of Irr(N ). With Theorem 4.3.1 we shall give a formal proof that (I) and (II), together with the condition D = 0, determine the character table T of G uniquely. We only observe here that in (II) we do not require the knowledge of the orbits of G on the set of conjugacy classes of N . This is due to the following general fact, which we mention without proof: if the character table of a normal subgroup N of G is given, together with the orbits of the action of G on Irr(N ) (whithout any further information about this action), then the orbits of G on the set of conjugacy classes of N can be uniquely determined; conversely,

4.2. VANISHING OF CHARACTER VALUES

61

the character table of N and the orbits of G on the set of conjugacy classes of N determine the orbits of G on Irr(N ). We conclude this section by noting that ingredient (II) can be further weakened (though it is sufficient as it stands for our purposes), because, instead of the character table of N we rather used the table T˜, which displays the values of the sums over the G-orbits of Irr(N ). We shall come back to this remark in Section 4.4.

4.2

Vanishing of character values

The matter of this section is the vanishing of the submatrix D of the character table T of G, as described above. The following two lemmas show how the vanishing of a column (respectively row) of D is equivalent to a conditon on the conjugacy class (respectively irreducible character) of G which corresponds to that column (respectively row). Since these results cannot be easily found in the literature in this form, we shall give their proofs in full. Lemma 4.2.1 Let N be a normal subgroup of the group G, let g ∈ G and let π : G → G/N be the natural epimorphism. Then any two of the following conditions are equivalent: (a) χ(g) = 0 for all χ ∈ Irr(G) \ Irr(G/N ); (b) |CG (g)| = |CG/N (g π )|; (c) g G = g G · N ; (d) gN ⊆ g G ; (e) N ⊆ bg, Gc. Proof ((a) ⇐⇒ (b)) By using the second orthogonality relation we get X

|CG/N (g π )| =

χ∈Irr(G/N )

|χ(g)|2 ≤

X

|χ(g)|2 = |CG (g)|,

χ∈Irr(G)

with equality if and only if χ(g) = 0 for all χ ∈ Irr(G) \ Irr(G/N ). π ((b) ⇐⇒ (c)) Since (g π )(x ) = (g x )π for all g, x ∈ G, we have π

((g π )(x ) )π

−1

= g x · N and ((g π )G/N )π

−1

= g G · N.

62


Therefore

−1

g G ⊆ g G · N = ((g π )G/N )π . It follows that |g G | ≤ |N | · |(g π )G/N )|, which is equivalent to |CG (g)| ≥ |CG/N (g π )|. We have equality here if and only if g G = g G · N . ((c) ⇐⇒ (d) ⇐⇒ (e)) This is easy.

¤

If N is a normal subgroup of the group G and θ ∈ Irr(N ), let us write Irr(G, θ) = {χ ∈ Irr(G)|[χN , θ] > 0}, where the brackets [ , ] denote the scalar product of characters. Since [χN , θ] = [χ, θG ] by Frobenius Reciprocity, Irr(G, θ) is the set of all irreducible constituents of θG . Lemma 4.2.2 Let N C G, let θ ∈ Irr(N ) and χ ∈ Irr(G, θ). Let e = [χN , θ] and let t be the number of distinct G-conjugates of θ. Then the following conditions are equivalent: (a) Irr(G, θ) = {χ}; (b) χ(g) = 0 for all g ∈ G \ N ; (c) e2 t = |G : N |. Proof Let θ = θ1 , . . . , θt be all the distinct G-conjugates of θ; then according to Clifford’s Theorem we have χN = e

t X

θi , where e = [χN , θ] = [χ, θG ].

i=1

Thus χ is an irreducible constituent of θG with multiplicity e. It follows that Irr(G, θ) = {χ} if and only if θG = eχ, or equivalently θG (1) = eχ(1). But θG (1) = |G : N |θ(1) and χ(1) = etθ(1), and so Irr(G, θ) = {χ} is equivalent to |G : N | = e2 t. According to [13, Lemma (2.29)] we have e2 t = [χN , χN ] ≤ |G : N |[χ, χ] = |G : N |,

4.3. A TOOL FOR COMPARING CHARACTER TABLES

63

and equality holds if and only if χ(g) = 0 for all g ∈ G \ N . This concludes the proof. ¤ It follows from Lemma 4.2.1 that the obvious character-theoretic expression of the condition D = 0, namely χ(g) = 0 for all g ∈ G \ N and for all χ ∈ Irr(G) \ Irr(G/N ), has a purely group-theoretic equivalent, namely |CG (g)| = |CG/N (gN )| for all g ∈ G \ N. This condition on G, with respect to a normal subgroup N , has been given a name. Definition 4.2.3 Let N be a proper non-trivial normal subgroup of the group G. The pair (G, N ) is called a Camina pair if the following condition holds: |CG (g)| = |CG/N (gN )| for all g ∈ G \ N. Camina pairs have been introduced in [2] as a generalization of Frobenius groups; in fact it is easy to see that (G, N ) is a Camina pair if G is a Frobenius group and N is its Frobenius kernel. Further examples of Camina pairs are given by (G, G0 ), where G is an extraspecial p-group. We refer to [2], [3] and [16] for series of results on Camina pairs. We observe that, according to Lemma 4.2.1, Camina pairs are also characterized by the following property: the inverse image of each non-trivial conjugacy class (gN )G/N of the factor group G/N , under the natural epimorphism π : G → G/N , is a conjugacy class of G, namely g G . (Let us note that in general the inverse image under π of a conjugacy class of G/N is a union of conjugacy classes of G.) Finally, let us mention a remarkable property of Camina pairs: if (G, N ) is a Camina pair, then every chief series of G must pass through N ; in other words, each normal subgroup of G, either contains N , or is contained in N .

4.3

A tool for comparing character tables

The following theorem provides a rigorous formulation of the result which we sketched in Section 4.1.

64


Theorem 4.3.1 Let N1 C G1 , N2 C G2 , and suppose that the following conditions hold: (i) G1 /N1 and G2 /N2 have identical character tables; (ii) N1 and N2 have identical character tables, via the bijections α ˜ : N1 → N2 and

β˜ : Irr(N1 ) → Irr(N2 ),

and, moreover, β˜ takes each G1 -orbit of Irr(N1 ) onto some G2 -orbit of Irr(N2 ); (iii) (G1 , N1 ) and (G2 , N2 ) are Camina pairs. Then G1 and G2 have identical character tables. Proof By hypothesis (i), there exist bijections α ¯ : G1 /N1 → G2 /N2 and

β¯ : Irr(G1 /N1 ) → Irr(G2 /N2 ),

such that ¯

χβ (xα¯ ) = χ(x) for all x ∈ G1 /N1 and for all χ ∈ Irr(G1 /N1 ). Let πi : Gi → Gi /Ni , for i = 1, 2, be the natural epimorphisms. Let α : G1 \ N1 → G2 \ N2 be any bijection making the following diagram commute: G1 \ N1   π1 y

α

−−−→

G2 \ N2   π2 y

α ¯

(G1 /N1 ) \ {1} −−−→ (G2 /N2 ) \ {1} Let us extend α to a bijection α : G1 → G2 by nα = nα˜ for all n ∈ N1 .


65

Then the extended α also satisfies απ2 = π1 α ¯. If χ ∈ Irr(Gi ) \ Irr(Gi /Ni ) (for i = 1 or 2), then hypothesis (iii) together with Lemma 4.2.1 guarantee that χ(g) = 0 for all g ∈ Gi \ Ni . Furthermore, if θ is an irreducible constituent of the restriction χNi , then Irr(Gi , θ) = {χ}, according to Lemma 4.2.2. Let us define a map: β : Irr(G1 ) \ Irr(G1 /N1 ) → Irr(G2 ) \ Irr(G2 /N2 ), as follows. If χ ∈ Irr(G1 ) \ Irr(G1 /N1 ), let θ be an irreducible constituent of χN1 , so that Irr(G1 , θ) = {χ} and θ is not the trivial character of N1 . ˜ ˜ It follows that θβ is not the trivial character of N2 , and hence Irr(G2 , θβ ) ˜ is a subset of Irr(G2 ) \ Irr(G2 /N2 ). Therefore Irr(G2 , θβ ) = {χ}, ˆ for some χˆ ∈ Irr(G2 ) \ Irr(G2 /N2 ) (namely χˆ is the unique irreducible constituent of ˜ the induced character (θβ )G2 ). Then define χβ = χ. ˆ This definition does not depend on the only choice we made, namely the choice of an irreducible constituent θ of χN1 , because β˜ takes G1 -conjugate characters of N1 to G2 -conjugate characters of N2 , according to hypothesis (ii). By Clifford’s Theorem and Lemma 4.2.2, the map which sends any χ ∈ Irr(Gi ) \ Irr(Gi /Ni ) (for i = 1 or 2) to the set of the irreducible constituents of χNi is a bijection from Irr(Gi ) \ Irr(Gi /Ni ) onto the set of Gi -orbits of Irr(Ni ) \ {1Ni }. Since β˜ is a bijection, it is clear that the map β is also a bijection. Let us extend β to a bijection β : Irr(G1 ) → Irr(G2 ) by defining ¯

χβ = χβ for all χ ∈ Irr(G1 /N1 ). It will follow that G1 and G2 have identical character tables, once we show that χβ (g α ) = χ(g) for all g ∈ G1 and for all χ ∈ Irr(G1 ).

66


We will prove this by distinguishing three cases. Case 1: χ ∈ Irr(G1 /N1 ), g ∈ G1 . We have ¯

¯

χβ (g α ) = χβ (g απ2 ) = χβ (g π1 α¯ ) = χ(g π1 ) = χ(g). Case 2: χ ∈ Irr(G1 ) \ Irr(G1 /N1 ), g ∈ N1 . Let θ be an irreducible constituent of χN1 and let θ = θ1 , · · · , θt be all the distinct G1 -conjugates of θ. Then χN1 = e

t X

θj ,

j=1

and the positive integer e is determined by e2 t = |G1 : N1 |, according to ˜ ˜ ˜ ˜ Lemma 4.2.2. The G2 -conjugates of θβ are θβ = θ1β , · · · , θtβ , by hypothesis (ii); hence (χβ )N2 = eˆ

t X β˜

θj ,

j=1

and eˆ2 t = |G2 : N2 |, by Lemma 4.2.2 again. But the groups G1 /N1 and G2 /N2 have identical character table, in particular they have the same order. This forces e and eˆ to be the same number, therefore (χβ )N2 = e

t X β˜

θj .

j=1

But then we have β

α

χ (g ) = e

t X β˜

α ˜

θj (g ) = e

j=1

t X

θj (g) = χ(g).

j=1

Case 3: χ ∈ Irr(G1 ) \ Irr(G1 /N1 ), g ∈ G1 \ N1 . Since χ and χβ vanish on G1 \ N1 and G2 \ N2 respectively, we have χβ (g α ) = 0 = χ(g). This completes the proof.

¤


67

The previous theorem will actually be used in presence of much stronger hypotheses, namely those of the next corollary. If N is an abelian normal subgroup of the group G, we can regard N as a ZG-module, with G acting on N by conjugation. Since N is contained in the kernel of this action, N can also be regarded as a Z(G/N )-module. If H is a group and γ : H → G/N is a group homomorphism, then N becomes an ZH-module in the usual way. Corollary 4.3.2 Let Ni be an abelian normal subgroup of Gi , for i = 1, 2, and suppose that the following conditions hold: (i) there exists a group isomorphism α ¯ : G1 /N1 → G2 /N2 ; (ii) there exists a Z(G1 /N1 )-module isomorphism α ˜ : N1 → N2 , where the Z(G2 /N2 )-module N2 is regarded as a Z(G1 /N1 )-module via the isomorphism α ¯; (iii) (G1 , N1 ) and (G2 , N2 ) are Camina pairs. Then G1 and G2 have identical character tables. Proof Suppose that the hypotheses of the corollary hold. Then hypotheses (i) and (iii) of Theorem 4.3.1 are clearly satisfied. Let us define a map β˜ : Irr(N1 ) → Irr(N2 ) by means of the formula ˜

−1

θβ (n) = θ(nα˜ ) for all n ∈ N2 and for all θ ∈ Irr(N1 ). ˜ Then N1 and N2 have identical character tables, via the bijections α ˜ and β. Let θ ∈ Irr(N1 ) and g ∈ G1 ; then, for all n ∈ N1 , we have ˜

−1

−1

−1

−1

−1

˜

−1

˜

(θg )β (n) = θg (nα˜ ) = θ((nα˜ )g ) = θ((ng )α˜ ) = θβ (ng ) = (θβ )g (n). Hence

˜

˜

(θg )β = (θβ )g for all θ ∈ Irr(N1 ) and for all g ∈ G1 .

68


It follows that β˜ takes each G1 -orbit of Irr(N1 ) onto some G2 -orbit of Irr(N2 ). Thus hypothesis (ii) of Theorem 4.3.1 is also satisfied, and the desired conclusion follows. ¤ Perhaps the simplest situation in which Corollary 4.3.2 applies is when G1 and G2 are extraspecial p-groups of the same order, and N1 , N2 are their centres; in fact (Gi , Ni ) is obviously a Camina pair in this case (for i = 1, 2). More generally, (G, Z(G)) is easily seen to be a Camina pair if G is a semi-extraspecial p-group, according to the following definition, which was introduced in [1]. Definition 4.3.3 A non-trivial p-group G is called semi-extraspecial if G/N is extraspecial for each maximal subgroup N of Z(G). A semi-extraspecial p-group G is obviously a special p-group, and |G : Z(G)| = p2n for some positive integer n. It turns out then that |Z(G)| ≤ pn (see [1]). For example, Suzuki 2-groups of type B, C, D (see [9] or [11, Chapter VIII, §7]) are semi-extraspecial and satisfy |G : Z(G)| = |Z(G)|2 . According to Corollary 4.3.2, the character table of a semi-extraspecial p-group G is completely determined by the two numbers |G : Z(G)| and |Z(G)|. Consequently, semiextraspecial p-groups provide plenty of examples of non-isomorphic groups which have the same character table.

4.4

A generalization

In this section we shall generalize Theorem 4.3.1 in two different directions. Our first generalization concernes a weakening of the condition that the pairs of groups (G1 , N1 ) and (G2 , N2 ) are Camina pairs, that is to say, hypothesis (iii) of Theorem 4.3.1. The condition that a group G forms a Camina pair, together with some normal subgroup N , is indeed quite a strong requirement on G, as it appears for instance from the results of [3] and [16]. The more general condition which we propose is better illustrated by using the language of Section 4.1.

4.4. A GENERALIZATION

69

Let us consider two proper non-trivial normal subgroups N and M of a group G, with N ≤ M . The character table T of G assumes the following form, after possibly rearranging the conjugacy classes and the irreducible characters of G:  Irr(G/M ) Irr(G/N ) \ Irr(G/M ) Irr(G) \ Irr(G/N )

N

M \N G\M



A11 A12 A13    A21 A22 A23  A31 A32 A33

We shall assume that A33 is the zero matrix (this clearly specializes to (G, N ) being a Camina pair, when N = M ). An (informal) argument similar to that used in Section 4.1 suggests that it should be possible to obtain T starting from the character table of the factor group G/N , and the character table of the normal subgroup M , together with the knowledge of the orbits of G on the set of conjugacy classes of M . It also appear that there must be some kind of compatibility between the character tables of G/N and M , where these two pieces of information overlap, namely on the normal section M/N of G. These heuristic considerations lead to the following theorem, which generalizes Theorem 4.3.1. Theorem 4.4.1 Let Ni C Gi and Mi C Gi for i = 1, 2, with Ni ≤ Mi , and suppose that the following conditions hold: (i) G1 /N1 and G2 /N2 have identical character tables, via the bijections α ¯ : G1 /N1 → G2 /N2 and

β¯ : Irr(G1 /N1 ) → Irr(G2 /N2 );

(ii) M1 and M2 have identical character tables, via the bijections α ˜ : M1 → M2 and

β˜ : Irr(M1 ) → Irr(M2 ),

and, moreover, β˜ takes each G1 -orbit of Irr(M1 ) onto a G2 -orbit of Irr(M2 );

70


(iii) g Gi = g Gi · Ni for all g ∈ Gi \ Mi , for i = 1, 2; (iv) (M1 /N1 )α¯ = M2 /N2 , and the diagram M1

  π1 y

α ˜

−−−→

M2

  π2 y

α ¯

M1 /N1 −−−→ M2 /N2 is commutative, where πi : Mi → Mi /Ni are the natural epimorphisms (for i = 1, 2). Then G1 and G2 have identical character tables. Proof A full proof can be given along the lines of the proof of Theorem 4.3.1, but we shall not give it here. Let us only notice that we need hypothesis (iv) here, in order to be able to extend the bijection α ˜ to a bijection α : G1 → G2 which satisfies απ2 = π1 α ¯. ¤ Now we propose a second generalization of Theorem 4.3.1, which weakens hypothesis (ii). Informally, as we anticipated at the end of Section 4.1, hypothesis (ii) of Theorem 4.3.1 can be replaced by the weaker requirement that the matrices T˜1 and T˜2 are equal, where the matrix T˜i is obtained from the character table of Ni as we did in Section 4.1, and thus T˜i displays the values of the sums of irreducible characters of Ni over Gi -orbits, as functions of the conjugacy classes of Gi contained in Ni . This weaker condition will be made precise in the following theorem. Theorem 4.4.2 Let N1 C G1 and N2 C G2 . For i = 1, 2, let Ii denote the set of the (possibly reducible) characters ψ of Ni such that ψ=

t X

θj

j=1

for some Gi -orbit θ1 , . . . , θt of Irr(Ni ) (where θ1 , . . . , θt are pairwise distinct). Suppose that the following conditions hold: (i) G1 /N1 and G2 /N2 have identical character tables;

4.4. A GENERALIZATION

71

(ii) there exist bijections α ˜ : N1 → N2 and

β˜0 : I1 → I2

such that ˜0

ψ β (nα˜ ) = ψ(n) for all n ∈ N1 and for all ψ ∈ I1 ; (iii) (G1 , N1 ) and (G2 , N2 ) are Camina pairs. Then G1 and G2 have identical character tables. Proof This theorem can be proved essentially in the same way as Theorem 4.3.1. The main difference is when one defines χβ for χ ∈ Irr(G1 )\Irr(G1 /N1 ). We shall briefly sketch this part of the proof. Let χ ∈ Irr(G1 ) \ Irr(G1 /N1 ). Let us choose an irreducible constituent θ of χN1 ; then we have Irr(G1 , θ) = {χ}. Now, if we put e = [χN1 , θ], then we have χN1 = eψ for a unique ψ ∈ I1 , namely the sum of all G1 -conjugates of ˜0 ˆ = {χ} θ. Let θˆ be an irreducible constituent of ψ β ; then we have Irr(G2 , θ) ˆ β for some χˆ ∈ Irr(G2 ) \ Irr(G2 /N2 ). Finally, let us define χ = χ. ˆ Another remark that should be made is that the number t of G1 -conjugates ˆ in fact, we have of θ equals the number tˆ of G2 -conjugates of θ; t = [ψ, ψ] =

1 X β˜0 α˜ 2 1 X ˜0 ˜0 |ψ(g)|2 = |ψ (g )| = [ψ β , ψ β ] = tˆ. |G1 | g∈G1 |G2 | g∈G2

For the rest of the proof, the reader is referred to the proof of Theorem 4.3.1. ¤ Clearly, hypothesis (ii) of Theorem 4.3.1 implies hypothesis (ii) of Theorem 4.4.2. However, we do not know of any example of groups G1 and G2 , with normal subgroups N1 and respectively N2 , which satisfy the hypotheses of Theorem 4.4.2 without satisfying the hypotheses of Theorem 4.3.1.

Chapter 5 Counterexamples 5.1

Introduction

In this chapter we shall construct pairs of groups (G, H) with identical character tables and derived length 2 and 3 respectively, which thus will be counterexamples to our Conjecture 3.1.1. Since our philosophy is that of trying to build our examples as small as possible we shall assume that G and H satisfy Hypotheses 3.1.2. With these hypotheses, a fairly detailed description of the structure of G and H is given in Chapter 3, in particular by Lemmas 3.2.1, 3.3.1 and 3.3.4. Thus, using for a moment a common notation for G and H, each of the groups G and H is a semidirect product of the form [D](W × Q), where • D (which stands for ‘derived subgroup’) is a p-group containing the unique minimal normal subgroup N of G (respectively H), • D/N is abelian, • W is an abelian p-group, • Q is a non-trivial cyclic p0 -group, • Q acts faithfully and irreducibly on N by conjugation, 72

5.1. INTRODUCTION

73

• W acts faithfully on D by conjugation, • all Q-composition factors of D are Q-isomorphic. Furthermore, if we fix a chief series of H going from 1 to D = H 0 , according to our analysis carried out in Section 3.4, commutation in H 0 gives rise to an FQ-module epimorphism γ¯ : V ∧ V → V in the unary case (that is to say, when r = s, where r and s are as defined in Section 3.4), or γ¯ : V ⊗ V → V in the binary case (when r < s), where V denotes the elementary abelian p-group H 00 regarded as a faithful irreducible Fp Q-module by conjugation. We shall not attempt to describe all pairs of groups (G, H) which satisfy Hypotheses 3.1.2. However, at least for p 6= 2, we shall construct examples (G, H) which satisfy Hypotheses 3.1.2, and which give rise to any prescribed Fp Q-epimorphism γ¯ : V ∧ V → V or γ¯ : V ⊗ V → V . In our examples, the subgroup D will have the smallest possible Q-length, namely 2 in the unary case, and 3 in the binary case. In all cases the factor group D/N will be elementary abelian. Now a word should be spent on W . We shall assume that [D, W ] ≤ N . In particular (DW )0 ≤ N (we shall have equality for H), and since N ≤ Z(DW ) because N is a minimal normal subgroup of G (respectively H), the subgroup DW will be nilpotent of class at most 2 (actually, exactly 2). According to Lemma 2.2.2 (with H1 /K1 = D/N , H2 /K2 = W and H3 /K3 = N ), the map ϕw : D/N → N x 7→ [x, w] is a Q-homomorphism for all w ∈ W , and the map γˆ : W → HomQ (D/N, N ) w 7→ ϕw is a group homomorphism. Actually γˆ is a monomorphism, because W acts faithfully on D. We shall employ Corollary 4.3.2 to prove that the groups G and H of our examples have identical character tables. In order to satisfy hypothesis (iii) of that corollary we shall require that the map γˆ above is surjective, and hence that it is a group isomorphism.

74

5.2

CHAPTER 5. COUNTEREXAMPLES

General construction

In the present section we shall develop the part of the construction which is common to the groups G and H of all the examples which we are going to build in this chapter and we shall prove that the resulting groups G and H have identical character tables. Let us fix a prime p and make the following assumptions (for i = 1, 2): (1) Di is a p-group with an elementary abelian subgroup Ni such that Φ(Di ) ≤ Ni ≤ Z(Di ); (2) Qi is a non-trivial cyclic p0 -group of automorphisms of Di which normalizes Ni and acts faithfully and irreducibly on Ni ; hence Ni becomes a faithful irreducible module for Qi over Fp ; (3) all Qi -composition factors of Di are Qi -isomorphic; (4) there is a group isomorphism σ : Q1 → Q2 ; (5) N2 regarded as an Fp Q1 -module via σ is isomorphic to the Fp Q1 -module N1 ; (6) D1 and D2 have the same order (in particular the Q1 -length of D1 equals the Q2 -length of D2 ). Starting from these ingredients, we shall define a certain group Wi of automorphisms of Di . Let us regard the elementary abelian p-group Di /Ni as an Fp Qi -module by conjugation. According to Maschke’s Theorem, the module Di /Ni is semisimple, because p does not divide |Qi |. If V denotes Ni regarded as an Fp Qi -module, then by assumption V is irreducible and all composition factors of Di /Ni as an Fp Qi -module are isomorphic to V . Hence Di /Ni is L isomorphic to lj=1 Vj , where V1 , . . . , Vl are copies of V , and the number l is the same for i = 1, 2, because the Q1 -length of D1 /N1 equals the Q2 -length of D2 /N2 . Now we have the Fp Qi -module isomorphism   l l M M HomFp Qi (Vj , V ) HomFp Qi  Vj , V  ∼ = j=1

j=1

5.2. GENERAL CONSTRUCTION

75

(which holds more generally if V1 , . . . , Vl , and V are arbitrary Fp Qi -modules). According to Theorem 2.3.1, the ring EndFp Qi (V ) is a field of pn elements, where |V | = pn . Since Vj ∼ = V , we have | HomFp Qi (Vj , V )| = |V | for all j = 1, . . . , l. It follows that | HomFp Qi (Di /Ni , Ni )| = |Di : Ni |. Later on we shall also need the following fact, which is easy to prove: for any element xNi of Di /Ni with x 6∈ Ni there exists some ϕ ∈ HomFp Qi (Di /Ni , Ni ) such that (xNi )ϕ 6= 1. Now we shall associate to each ϕ ∈ HomFp Qi (Di /Ni , Ni ) an automorphism wϕ of Di . Let ϕ : Di /Ni → Ni be an Fp Qi -homomorphism. Since Ni is a central subgroup of Di , the map wϕ : Di → Di x 7→ x(xNi )ϕ is a group automorphism of Di and it clearly commutes with the action of Qi . Let Wi be the set of all automorphisms of Di which arise in this way, in other words Wi = {wϕ | ϕ ∈ HomFp Qi (Di /Ni , Ni )}. Then Wi is a subgroup of AutFp Qi (Di ). In fact, the map δ : HomFp Qi (Di /Ni , Ni ) → AutQi (Di ) ϕ 7→ wϕ is a group homomorphism, because xwϕ1 +ϕ2 = x(xNi )ϕ1 +ϕ2 = x(xNi )ϕ1 (xNi )ϕ2 = (xwϕ1 )wϕ2 for all x ∈ Di , and Wi is its image. Clearly δ is a monomorphism, namely wϕ is the identity automorphism of Di exactly when ϕ is the zero homomorphism. It follows in particular that the order of Wi equals the order of HomFp Qi (Di /Ni , Ni ), which we computed earlier; hence |Wi | = |Di : Ni |.

76


We observe that the group Wi is an elementary abelian p-group, because HomFp Qi (Di /Ni , Ni ) is a vector space over Fp . Let us also notice that if ϕw , for w ∈ Wi , denotes the Fp Q-module homomorphism ϕw : Di /Ni → Ni xNi 7→ [x, w], then the group homomorphism γˆ : Wi → HomFp Qi (Di /Ni , Ni ) w 7→ ϕw , is an isomorphism, in accordance with the requirement which we made at the end of Section 5.1, and is the inverse of the isomorphism δ : HomFp Qi (Di /Ni , Ni ) → Wi defined above. The subgroups Wi and Qi of Aut(Di ) satisfy Wi ∩ Qi = [Wi , Qi ] = 1, hence Wi Qi is a subgroup of Aut(Di ), and is the internal direct product of Wi and Qi . Hence Wi Qi is canonically isomorphic to the external direct product Wi × Qi , which thus acts on Di . Let us define groups G1 and G2 as the semidirect products Gi = [Di ](Wi × Qi ). We observe that Ni is a minimal normal subgroup of Gi , because Qi acts irreducibly on Ni . It would not be difficult to prove that Ni is the unique minimal normal subgroup of Gi . However, according to the last observation of Section 4.2, this will also follow from the fact that (Gi , Ni ) is a Camina pair, which is proved in the following lemma. Lemma 5.2.1 Let G1 and G2 as above. Then (i) G01 = D1 and G02 = D2 ; (ii) G1 and G2 have identical character tables; (iii) (G1 , N1 ) and (G2 , N2 ) are Camina pairs.


77

Proof (i) Since [Di , Qi ] ≤ Di and [Di , Qi ] C hDi , Qi i = Di Qi , the group [Di , Qi ] is a normal subgroup of Di and is also normalized by Qi (in other words, [Di , Qi ] is a normal Qi -subgroup of Di ). Furthermore, Qi centralizes the factor group Di /[Di , Qi ]. Since by hypothesis all Qi -composition factors of Di are Qi -isomorphic and Qi does not centralize Ni , we have [Di , Qi ] = Di . In particular, it follows that G0i = Di , because Gi /Di is clearly abelian. (ii) We shall apply Corollary 4.3.2 to the groups Gi and the normal subgroups Ni . Let us check then that the hypotheses of Corollary 4.3.2 are satisfied. First of all, let us show that G1 /N1 ∼ = G2 /N2 . We assumed that Di is a Qi group all whose Qi -composition factors are Qi -isomorphic, for i = 1, 2. Let us regard the Q2 -group D2 as a Q1 -group via the isomorphism σ : Q1 → Q2 . It follows that all Q1 -composition factors of D2 are Q1 -isomorphic. Since we also assumed that N2 , viewed as an Fp Q1 -module, is isomorphic to N1 , we get that each Q1 -composition factor of D1 is Q1 -isomorphic to each Q1 -composition factor of D2 . Now the Q1 -factor groups D1 /N1 and D2 /N2 are elementary abelian p-groups; hence they can be regarded as Fp Q1 -modules, and they are semisimple according to Maschke’s theorem, because p does not divide |Q1 |. Also, they have the same Q1 -length, let us say l. Hence, if V denotes N1 regarded as an Fp Q1 -module, then each of D1 /N1 and D2 /N2 is isomorphic as an Fp Q1 -module to the direct sum of l copies of V , in particular D1 /N1 and D2 /N2 are isomorphic as Fp Q1 -modules. Let us fix an Fp Q1 -isomorphism τ : D1 /N1 → D2 /N2 . In other words, let τ be a group isomorphism which satisfies ((xN1 )ξ )τ = ((xN1 )τ )ξ

σ

for all x ∈ D1 and for all ξ ∈ Q1 .

Now, the groups W1 and W2 are elementary abelian p-groups of the same order; in fact, |W1 | = |D1 /N1 | = |D2 /N2 | = |W2 |. Let us fix a group isomorphism ρ : W1 → W2 . Since Wi centralizes Qi and Di /Ni by construction, we have Gi /Ni = (Di Wi Qi )/Ni ∼ = (Di Qi /Ni ) × Wi .

78


Let α ¯ : G1 /N1 → G2 /N2 be the map such that (xwξN1 )α¯ = (xN1 )τ wρ ξ σ for all x ∈ D1 , for all w ∈ W1 and for all ξ ∈ Q1 . It is clear that α ¯ is a bijection; moreover, α ¯ is a group isomorphism. In fact, we have −1 ¯ 1 ))α¯ = ((x¯ ¯ 1 )α¯ ((xwξN1 )(¯ xw¯ ξN xξ )(ww)(ξ ¯ ξ)N −1 ¯σ = (x¯ xξ N1 )τ (ww) ¯ ρ (ξ ξ)

wρ w¯ ρ ξ σ ξ¯σ = ((xN1 )τ wρ ξ σ )((¯ xN1 )τ w¯ ρ ξ¯σ ) ¯ 1 )α¯ . = (xwξN1 )α¯ (¯ xw¯ ξN

= (xN1 )τ ((¯ xN1 )τ )(ξ

σ )−1

Thus hypothesis (i) of Corollary 4.3.2 has been verified. Let us regard Ni as an Fp Qi -module by conjugation. We can also regard N2 as an Fp Q1 -module via the isomorphism σ : Q1 → Q2 . We assumed that N1 and N2 are isomorphic as Fp Q1 -modules. Let α ˜ : N 1 → N2 be an Fp Q1 -module isomorphism. Now N1 can be regarded as an Fp (G1 /N1 )module by conjugation, while N2 can be regarded as an Fp (G2 /N2 )-module by conjugation and also as an Fp (G1 /N1 )-module via the isomorphism α ¯ : G1 /N1 → G2 /N2 which we defined earlier. Because Ni ≤ Z(Di Wi ), the Fp (G1 /N1 )-modules N1 and N2 can also be viewed as Fp (G1 /D1 W1 )-modules. Since the isomorphism α ¯ extends the isomorphism σ : Q1 → Q2 , and Qi (for i = 1, 2) is a complement for Di Wi in Gi , it is clear that the Fp Q1 -module isomorphism α ˜ : N1 → N2 is actually an Fp (G1 /D1 W1 )-module isomorphism and thus an Fp (G1 /N1 )-module isomorphism. This is exactly what is required by hypothesis (ii) of Corollary 4.3.2. It remains to chech that hypothesis (iii) of Corollary 4.3.2 is satisfied, namely that (G1 , N1 ) and (G2 , N2 ) are Camina pairs, or in other words, that giG = giG · Ni for all g ∈ Gi \ Ni , for i = 1, 2. According to Lemma 4.2.1, this is equivalent to Ni ⊆ bg, Gi c for all g ∈ Gi \ Ni .


79

We will distinguish three cases. Case 1: g ∈ Ni Wi \ Ni . Let us write g = xwϕ , with x ∈ Ni and wϕ ∈ W . Since wϕ 6= 1, the Fp Qi module homomorphism ϕ : Di /Ni → Ni is not the zero homomorphism; therefore ϕ is surjective, because Ni is an irreducible Fp Qi -module. Thus we have bDi , gc = bDi , wϕ c = (Di /Ni )ϕ = Ni . In particular Ni = bg, Di c ⊆ bg, Gi c. Case 2: g ∈ Di Wi \ Ni Wi . Let us write g = xw, with x ∈ Di \ Ni and w ∈ Wi . Then bg, Wi c = bx, Wi c = {(xNi )ϕ | ϕ ∈ HomFp Qi (Di /Ni , Ni )} is clearly a subgroup of Ni normalized by Qi , in other words an Fp Qi submodule of the irreducible Fp Qi -module Ni . As we remarked earlier, since xNi 6= Ni , there exists some ϕ ∈ HomFp Qi (Di /Ni , Ni ) such that (xNi )ϕ 6= 1. Hence bg, Wi c is not the trivial Fp Qi -submodule of Ni and thus we have bg, Wi c = Ni . In particular Ni ⊆ bg, Gi c. Case 3: g ∈ Gi \ Di Wi . Let us write g = xwξ, with x ∈ Di , w ∈ Wi and ξ ∈ Qi with ξ 6= 1. Then bNi , gc = bNi , ξc is an Fp Qi -submodule of Ni ; in fact, it is the image of the Fp Qi -module homomorphism N i → Ni x 7→ [x, ξ]. Since Qi acts faithfully on Ni and ξ 6= 1, we have bNi , gc 6= 1 and thus bNi , gc = Ni . In particular, Ni = bg, Ni c ⊆ bg, Gi c. Hence we have proved that giG = giG · Ni for all g ∈ Gi \ Ni , and thus hypothesis (iii) of Corollary 4.3.2 has also been verified. Its conclusion that G1 and G2 have identical character tables now follows. (iii) The fact that (G1 , N1 ) and (G2 , N2 ) are Camina pairs has been proved above. ¤

80


5.3

The unary case

This section and Section 5.5 are devoted to the construction of pairs (G, H) of groups, according to the pattern described in the previous section (with G1 = G and G2 = H). The derived subgroups D1 (abelian) and D2 (metabelian) will have Q1 -length and respectively Q2 -length 2 in this section and 3 in Section 5.5. As we said earlier, any Fp Q-module epimorphism γ¯ : V ∧ V → V or γ¯ : V ⊗ V → V, where V is a faithful irreducible Fp Q-module, can arise from concrete examples (G, H) satisfying Hypotheses 3.1.2, by means of the procedure described in Section 3.4. However, for the sake of simplicity we shall prove this fact only for p 6= 2 and limit ourselves to giving some representative examples for p = 2 (in Sections 5.4 and 5.6). Hence let us assume that p is an odd prime, and let us make the following assumptions: • Q is a (non-trivial) cyclic group of p0 -order; • V is a faithful irreducible Fp Q-module; • γ¯ : V ∧ V → V is a fixed Fp Q-module epimorphism. We observe that the dimension n of V over Fp is uniquely determined by p and |Q| according to Lemma 2.3.1. Let A = V ⊕ V be the direct sum of two copies of V . The Fp Q-factor module A/(0 ⊕ V ) and the Fp Q-submodule 0 ⊕ V are both isomorphic to V , in particular HomFp Q (A/(0 ⊕ V ), 0 ⊕ V ) ∼ = Fpn , as vector spaces over Fp , where n is the dimension of V over Fp . For any ϕ ∈ HomFp Q (A/(0 ⊕ V ), 0 ⊕ V ), the map wϕ defined by awϕ = a + (a + (0 ⊕ V ))ϕ for all a ∈ A,

5.3. THE UNARY CASE

81

is an automorphism of A as an Fp Q-module. Then W = {wϕ | ϕ ∈ HomFp Q (A/(0 ⊕ V ), 0 ⊕ V )} is a subgroup of AutFp Q (A). The order of W equals the order of V , namely pn . In fact, if we put D1 = A, N1 = 0 ⊕ V , Q1 = Q and use a multiplicative notation for D1 , then D1 satisfies conditions (1), . . . , (6) of Section 5.2; hence W is exactly the subgroup W1 of AutQ1 (D1 ) which is defined there and has order |D1 : N1 | = pn . Let us define G1 = [D1 ](W1 × Q1 ) as in Section 5.2. The group G1 will also be called G here and we shall therefore write G = [A](W × Q). According to Lemma 5.2.1, we have G0 = A and thus G00 = 1, that is to say, G is metabelian. Let us pass to the construction of the group H. Let Fˆ be a free group of rank n. Then F = Fˆ /γ3 (Fˆ )Fˆ p is a free nilpotent group of class two and exponent p (that is to say, F is a free object in the variety of nilpotent groups of class 2 and exponent p, see [18]). If p were the prime 2, then the group F would be abelian, but since we assumed p 6= 2, we have F 0 6= 1. More precisely, F 0 is elementary abelian of order pn(n−1)/2 . In fact, according to Lemma 2.6.3, if F = hx1 , . . . , xn i, then a basis of F 0 over Fp is given by the set of basic commutators of weight 2 on x1 , . . . xn , namely {[xj , xk ] | 1 ≤ k < j ≤ n}. The factor group F/F 0 is a free abelian group of exponent p and rank n, in other words it is elementary abelian of order pn . Let us fix an Fp -linear isomorphism τ : V → F/F 0

82


and let us make F/F 0 into an Fp Q-module isomorphic to V via τ , namely let us define an action of Q on F/F 0 according to the formula −1

(xF 0 )ξ = ((xF 0 )τ ξ)τ for all x ∈ F and for all ξ ∈ Q, and extend it Fp -linearly to an action of Fp Q on F/F 0 . Thus F/F 0 becomes an Fp Q-module and τ an Fp Q-module isomorphism. Let us fix a generator ξ of Q. The automorphism of F/F 0 induced by ξ can be lifted to an automorphism ξˆ of F , because F is a relatively free group (see [18]). Since F is a finite group, the automorphism ξˆ has finite order; this is certainly a multiple of |Q|, which is the order of ξ. We may assume that the order of ξˆ is not a multiple of p, otherwise we can replace ξˆ with a suitable nt power ξˆp (t integer), which induces on F/F 0 the same automorphism as ξˆ does. Now ξˆ|Q| is an automorphism of p0 -order of the p-group F , which induces the identity automorphism on F/Φ(F ) = F/F 0 . According to [10, Kapitel III, Satz 3.18] then ξˆ|Q| is the identity automorphism of F , and thus ξˆ has order |Q|. Therefore F can be regarded as a Q-group. In particular F 0 can be viewed as a (semisimple) Fp Q-module. According to Lemma 2.2.1 (with H1 /K1 = H2 /K2 = F/F 0 and H3 /K3 = F 0 ), commutation in F gives rise to an Fp -bilinear map δ : (F/F 0 ) × (F/F 0 ) → F 0 (xF 0 , yF 0 ) 7→ [x, y]. Since δ is skew-symmetric and F 0 is a Q-subgroup of the Q-group F , we obtain in turn an Fp Q-module homomorphism ¯δ : (F/F 0 ) ∧ (F/F 0 ) → F 0 (xF 0 ) ∧ (yF 0 ) 7→ [x, y]. Because we assumed that p is odd, ¯δ is an isomorphism. In fact, the set {(xj F 0 ) ∧ (xk F 0 ) | 1 ≤ k < j ≤ n} is a basis of (F/F 0 ) ∧ (F/F 0 ) over Fp , and ¯δ sends it bijectively onto the set of basic commutators of weight 2 on x1 , . . . , xn , which is an Fp -basis of F 0 , as we saw earlier. The Fp Q-module isomorphism τ : V → F/F 0

5.3. THE UNARY CASE

83

induces an Fp Q-module isomorphism τ ∧ τ : V ∧ V → (F/F 0 ) ∧ (F/F 0 ) in an obvious way. We get the following commutative diagram of Fp Q-module homomorphisms: ¯ δ

(F/F 0 ) ∧ (F/F 0 ) −−−→ F 0 x  τ ∧τ 

V ∧V

  −1 ¯δ (τ ∧τ )−1 γ¯ y

¯ γ

−−−→ V

Now let us put −1 ¯ K = ker (¯δ (τ ∧ τ )−1 γ¯ ) = (ker γ¯ )(τ ∧τ )δ .

The factor group F 0 /K is then an Fp Q-module, and the Fp Q-module epimorphism ¯δ −1 (τ ∧ τ )−1 γ¯ : F 0 → V induces an Fp Q-module isomorphism ν : F 0 /K → V −1 such that ¯δ (τ ∧ τ )−1 γ¯ = πν, where π : F 0 → F 0 /K is the natural epimorphism. Since K is also a central subgroup of F , we can form the factor group X = F/K, which is a p-group of class 2, exponent p and order p2n . ˆ having order |Q|) Let ξ¯ denote the automorphism of X (like ξ and ξ, ¯ be the subgroup of Aut(X) induced by the automorphism ξˆ of F and let Q ¯ ¯ ¯ generated by ξ. Let σ : Q → Q be the group isomorphism such that ξ σ = ξ. ¯ ¯ Then the Q-group X becomes a Q-group via σ −1 , it has Q-length 2 and 0 0 0 0 ¯ its Q-composition factors X/X = F/F and X = F /K, regarded as Fp Qmodules via σ, are both isomorphic to V . We clearly have Φ(X) = X 0 , and ¯ also Z(X) = X 0 , because Z(X) is a proper Q-subgroup of X which contains 0 0 ¯ the derived subgroup X , and X/X is an irreducible Fp Q-module. For each ϕ¯ ∈ HomFp Q¯ (X/X 0 , X 0 ),

the map w ¯ϕ¯ defined by xw¯ϕ¯ = x(xX 0 )ϕ¯ for all x ∈ X,

84


¯ The set is an automorphism of X commuting with ξ. ¯ = {w¯ϕ¯ | ϕ¯ ∈ HomF Q¯ (X/X 0 , X 0 )} W p ¯ equals the order of X/X 0 , is a subgroup of AutQ¯ (X), and the order of W ¯ then D2 satisfies namely pn . In fact, if we put D2 = X, N2 = X 0 and Q2 = Q, ¯ conditions (1), . . . , (6) of Section 5.2, and W is exactly the subgroup W2 of AutQ2 (D2 ) which is defined there. Let us define G2 = [D2 ](W2 × Q2 ) as in Section 5.2. The group G2 will also be called H here and we shall therefore write ¯ × Q). ¯ H = [X](W Since all assumptions of the previous section are satisfied, Lemma 5.2.1 applies and yields that G and H have identical character tables, and that H 0 = X, whence H 00 = X 0 and H 000 = 1. Thus G and H have identical character tables and G is metabelian, as we saw earlier, while H has derived length 3. ¯ We observe that the group X = H 0 has a unique Q-composition series, namely 1 < X 0 < X, which is also part of a chief series of H going from 1 to X. According to the analysis of commutation in X which we carried out in Section 3.4, the ¯ Fp Q-module epimorphism associated with our chief series is the following: ¯δπ : (X/X 0 ) ∧ (X/X 0 ) → X 0 (xX 0 ) ∧ (yX 0 ) 7→ [x, y]. ¯ Now, the Fp Q-module epimorphism ¯δπ coincides with our prescribed Fp Qmodule epimorphism γ¯ : V ∧ V → V , after identifying X/X 0 and X 0 with V by means of suitable Fp Q-module isomorphisms. In fact, one can easily check that the following diagram is commutative: ¯ δπ

(X/X 0 ) ∧ (X/X 0 ) −−−→ X 0 x  τ ∧τ 

V ∧V

  ν y

¯ γ

−−−→ V.

5.3. THE UNARY CASE

85

As we promised, under the assumption p 6= 2, we have constructed pairs of groups (G, H) which satisfy Hypotheses 3.1.2 and which give rise,through the method of Section 3.4, to any arbitrarily given Fp Q-module epimorphism γ¯ : V ∧ V → V . Let us remark that the normal Sylow p-subgroups of G and H, namely ¯ , have identical character tables, as one could easily P = AW and P¯ = X W show by applying Corollary 4.3.2. Of course P and P¯ are both metabelian, because they are nilpotent groups of class two, but we want to stress here that they are not isomorphic. Let us prove this fact. Let AP (and respectively AP¯ ) denote the set of the abelian subgroups of P (resp. P¯ ) of order p2n and containing P 0 (resp. P¯ 0 ). Once we prove that the sets AP and AP¯ have different cardinality, it will follow that P and P¯ are not isomorphic. First, let K be a non-trivial subgroup of Q, and let S be a subgroup of P of order p2n which contains P 0 and is normalized by K. According to Lemma 2.1.1 we have S = [S, K]CS (K), with [S, K] ≤ G0 = A and CS (K) ≤ CP (K) = W (this last assertion follows easily from statements (ii) and (iii) of Lemma 3.2.1). Let us assume that S is different from A and P 0 W , that is to say, [S, K] > P 0 and CS (K) 6= 1. Let w be a non-identity element of CS (K), and hence in particular of W ; then the map A → P0 a 7→ [a, w] is by construction a group epimorphism with kernel P 0 , and hence its restriction to the subgroup [S, K] of A is not the zero homomorphism. As a consequence, S is not abelian. It follows that A (which is G0 ) and P 0 W are the only abelian subgroups of P of order p2n which contain P 0 and are normalized by some non-trivial subgroup K of Q. Since Q acts on AP and every non-trivial subgroup K of Q fixes only the elements A and P 0 W of AP , it follows that all orbits of Q on AP \ {A, P 0 W } have length |Q|; therefore we have |AP | ≡ 2 mod |Q|. ¯ = X 0W ¯ is the only abelian subgroup of P¯ Similarly one proves that P¯ 0 W 2n 0 0 of order p which contains P¯ = X and is normalized by some non-trivial

86


¯ (because in this case H 0 = X is not abelian). Here Q ¯ acts on subgroup of Q 0 ¯ ¯ ¯ AP¯ , and every non-trivial subgroup of Q fixes only the element P W of AP¯ . It follows that ¯ |AP¯ | ≡ 1 mod |Q|. ¯ we conclude that P and P¯ are not isomorphic. Since |Q| = |Q|, Finally let us compute the smallest possible order of G (and hence of H) for a pair of groups (G, H) constructed by the above method. As we saw in Section 3.5, V cannot appear as a composition factor of V ∧ V unless the dimension of V over Fp is at least 4. Moreover, there exists a faithful irreducible module V of dimension 4 over Fp for a (non-trivial) cyclic p0 -group Q, such that V is isomorphic to a composition factor of V ∧ V , if and only if Q has order 5 and p ≡ 2 or 3 mod 5. Since in the present section we assumed that p is odd, we see that the smallest possible value for |G| occurs when p = 3 and |Q| = 5, in which case we have |G| = 312 · 5.

5.4

Matrix representations for the unary case

All groups G and H which we have constructed have natural faithful representations as groups of matrices over Fpn ; in other words, G and H are isomorphic to certain subgroups of GLr (pn ) for some r. Whereas G is isomorphic to a subgroup of GL3 (pn ), we need bigger matrices for representing H. The smallest possible size depends on the choice of the particular Fp Qmodule epimorphism γ¯ : V ∧ V → V . We shall not discuss such matrix representations in general. However, for each choice of an odd prime p and of a faithful irreducible module V for a cyclic p0 -group Q over Fp , such that V is a composition factor of V ∧ V , we shall choose a particular Fp Q-module epimorphism γ¯ : V ∧ V → V and give an explicit representation of the resulting group H as a subgroup of GL4 (pn ). We recall that if V is a faithful irreducible module for a cyclic group Q over Fp , then V is isomorphic to the module Vε defined in Theorem 2.3.1, for a suitable primitive |Q|th root of unity ε in Fpn , where n is the dimension of V over Fp . Furthermore, according to Lemma 3.5.1, the fact that V is a composition factor of V ∧ V (or of V ⊗ V ) depends only on the prime p and the order q of Q. As a consequence, we do not lose in generality if we take,

5.4. MATRIX REPRESENTATIONS FOR THE UNARY CASE

87

as the basic ingredients of our construction, an odd prime p and a positive integer q satisfying the conditions stated below. Let us fix an odd prime p and a positive integer q, such that pi + 1 ≡ pj mod q for some integers i, j, with i not divisible by n, where n is the multiplicative order of p mod q. It follows from the observation which precedes Lemma 3.5.1 that we may always assume that 0 < i ≤ n/2

and

0 ≤ j < n.

Even with this assumption, the pair (i, j) is in general not unique. For instance, if q is also a prime and p has multiplicative order q − 1 mod q (examples are given by (p, q) = (3, 5), (5, 7), (7, 11)), then {p, p2 , . . . , pq−2 } ≡ {2, 3, . . . , q − 1} mod q; consequently, if q ≥ 5 there exists at least one pair (i, j) of integers such that pi + 1 ≡ pj mod q, and with 0 < i ≤ (q − 1)/2

and

0 ≤ j < q − 1;

if q ≥ 7, there exist at least two such pairs. Let Q = hξi be a cyclic group of order q, and let ε be a primitive qth root of unity in Fpn . According to Theorem 2.3.1, the module Vε is a faithful irreducible module for Q over Fp , where the underlying vector space of Vε is the field Fpn , and the action of Q on Vε is given by vξ = εv for all v ∈ Vε . We shall also consider the Fp Q-module Vεpj defined similarly, which is isomorphic to Vε via the Fp Q-module isomorphism Vε → Vεpj j

v 7→ v p . Because of our assumptions on p and q, and according to Lemma 3.5.1, the Fp Q-module Vε ∧ Vε has a composition factor isomorphic to Vε , and hence

88


to Vεpj . Actually, we can explicitly define an Fp Q-epimorphism from Vε ∧ Vε onto Vεpj , as follows. Let γ¯ be the map γ¯ : Vε ∧ Vε → Vεpj such that

¯

i

i

(x ∧ y)γ¯ = xy p − xp y for all x, y ∈ Vε . i

i

Because the expression xy p −xp y is Fp -bilinear and skew-symmetric in (x, y), the map γ¯ is well defined and Fp -linear. We have ¯

¯

((x ∧ y)ξ)γ¯ = ((εx) ∧ (εy))γ¯ i

i

i

= ε1+p (xy p − xp y) ¯

i

= ε1+p (x ∧ y)γ¯ ¯

j

= εp (x ∧ y)γ¯ ¯

= (x ∧ y)γ¯ ξ, and hence it follows by Fp -linearity that γ¯ is an Fp Q-module homomorphism. Furthermore, γ¯ is surjective. In fact, a non-zero element of Vε ∧ Vε of the form x ∧ y (which of course is not a generic element of Vε ∧ Vε ) belongs to the kernel of γ¯ exactly when i

(xy −1 )p = xy −1 ; i

on the other hand, the field automorphism of Fpn given by x 7→ xp is not the identity automorphism, because i is not a multiple of n. It follows that the kernel of γ¯ is not the whole of Vε ∧Vε . Hence γ¯ is not the zero homomorphism, and thus it is an Fp Q-module epimorphism, because Vεpj is irreducible. (It should be said that not all Fp Q-module epimorphisms from V ∧ V onto V can be put into this particularly simple form by suitably identifying V with Vε and Vεpj .) With this particular choice of the Fp Q-module epimorphism γ¯ , let us define the group X = F/K as we did in Section 5.3. The set X3 whose elements are the matrices   1 a b i    1 ap  1


89 i

with a, b ∈ Fpn is clearly a subgroup of GL3 (pn ) (because the map a 7→ ap is a field automorphism of Fpn , and in particular it is Fp -linear). Furthermore, X3 is isomorphic to X. In fact, it is not difficult to define an epimorphism from F onto X with kernel K, after noticing that 

−1 

1 a b i    1 ap  1

−1    1 a ¯ ¯b 1 a b 1 a ¯ ¯b i  i  i     1 a ¯p   1 ap   1 a ¯p  1 1 1  

i

=

i

a¯ ap − ap a ¯

1 1

  .

1 It is also clear that the diagonal matrix 

ξ¯3 =  



1

 

ε ε

1+pi

normalizes X3 , and induces by conjugation an automorphism of X3 which corresponds to the automorphism ξ¯ of X defined in Section 5.3 (for a suitable ¯ choice of the isomorphism from X to X3 ). Thus the normal subgroup X Q of H is isomorphic to a subgroup of GL3 (pn ). Let us note in passing that X3 would be abelian if i were a multiple of n, which it is not in our case. Now we are ready to embed the whole of H into GL4 (pn ). First of all, we observe that the set X4 of the matrices 

j

1 a ap   1   1 



b i  ap    

1 with a, b ∈ Fpn is a subgroup of GL4 (pn ) isomorphic to X3 (and hence to X); in fact, the map 

1 a ap 1 a b   i  1   1 ap  7→   1  1 



j



b i  ap    

1

90


is an isomorphism from X3 onto X4 . The diagonal matrix  



1

    

ε

 ξ¯4 =  

j

εp



j

εp i

j

normalizes X4 (remember here that ε1+p = εp ). The automorphism of X4 induced by ξ¯4 by conjugation corresponds to the automorphism of X3 induced by ξ¯3 by conjugation (with respect to the given isomorphism from X3 onto X4 ) and thus to the automorphism ξ¯ of X (with respect to a suitable isomorphism from X onto X4 ). ¯ 4 of GL4 (pn ) consisting of the matrices Now the subgroup W     



1 1 1

   c 

1 with c ∈ Fpn is elementary abelian of order pn , centralizes ξ¯4 , and normalizes X4 ; in fact,     

1 1 1

−1  1       c   

j

a ap 1 1

1





 

   c 

b 1  pi   1 a 

1

1

1



j

1 a ap   1 =  1 

j



b + ap c  i  ap 

. 

1 ¯ × Q) ¯ is isomorphic to the subgroup Now it is clear that H = [X](W n ¯ 4 hξ¯4 i of GL4 (p ), namely the group of the matrices H4 = X4 W 

j

1 a ap   εl  j  εlp 

b i ap c j εlp

     


91

with a, b, c ∈ Fpn and l = 1, . . . , |Q|. Thus we have constructed a faithful matrix representation of H. It is much easier to give a matrix representation of the group G. In fact, it is straightforward to check that G is isomorphic to the subgroup G3 of GL3 (pn ) which consists of the matrices 



1 a  εl 

b c   l ε

with a, b, c ∈ Fpn and l = 1, . . . , |Q|; in particular, the normal Sylow psubgroup AW of G is isomorphic to a Sylow p-subgroup of GL3 (pn ). Let us also notice that G is isomorphic to the subgroup G4 of GL4 (pn ) which consists of the matrices 

j

1 a ap   εl  j  εlp 

b c lpj

     

ε

with a, b, c ∈ Fpn and l = 1, . . . , |Q|. In fact, an isomorphism from G3 onto G4 is given by the map 

j

1 a ap  1 a b  εl  εl c    7→  j  εlp  l ε 



j

bp

j

cp j εlp

   .  

Since G4 and H4 normalize each other, G4 H4 is a subgroup of GL4 (pn ). Clearly G4 H4 consists of the matrices 

j

1 a ap   εl  j  εlp 

b d c lpj ε

     

with a, b, c, d ∈ Fpn and l = 1, . . . , |Q|. The groups G4 and H4 are normal subgroups of G4 H4 of index pn . In particular, we have shown that our groups

92


G and H can be simultaneously embedded as normal subgroups into a group of order pn |G|. As a final remark, we observe that the construction of all groups of matrices of this section works as well if we drop our assumption that the prime p is odd. The resulting groups G4 and H4 do have identical character tables and derived length 2 and 3 respectively, even if p = 2. The only differences for p = 2 are that they do not correspond to any of the groups defined in Section 5.3, and that the normal Sylow 2-subgroups of G4 and H4 have exponent 4 = p2 instead of p. However, the construction of Section 5.3 could be easily modified in order to include the case p = 2. If we allow p = 2, then the smallest possible order for G4 and H4 drops down to 212 · 5.

5.5

The binary case

Let p be an odd prime, and let us make the following assumptions: • Q is a non-trivial cyclic group of p0 -order; • V is a faithful irreducible Fp Q-module; • γ¯ : V ⊗ V → V is a fixed Fp Q-module epimorphism. Let A = V ⊕ V ⊕ V be the direct sum of three copies of V . We have HomFp Q (A/(0 ⊕ 0 ⊕ V ), 0 ⊕ 0 ⊕ V ) ∼ = Fpn ⊕ Fpn , as vector spaces over Fp , where n is the dimension of V over Fp . For any ϕ ∈ HomFp Q (A/(0 ⊕ 0 ⊕ V ), 0 ⊕ 0 ⊕ V ), the map wϕ defined by awϕ = a + (a + (0 ⊕ 0 ⊕ V ))ϕ for all a ∈ A, is an automorphism of A as an Fp Q-module. The group W = {wϕ | ϕ ∈ HomFp Q (A/(0 ⊕ 0 ⊕ V ), 0 ⊕ 0 ⊕ V )}

5.5. THE BINARY CASE

93

is a subgroup of AutFp Q (A), of order |V ⊕ V | = p2n . With the notation of Section 5.2, we may take D1 = A, N1 = 0 ⊕ 0 ⊕ V , Q1 = Q, and thus obtain W1 = W . Let us construct the semidirect product G1 = [D1 ](W1 × Q1 ) as in Section 5.2. The group G1 will also be called G here and we shall therefore write G = [A](W × Q). According to Lemma 5.2.1, we have G0 = A; therefore G00 = 1, and thus G is metabelian. Let us pass to the construction of the group H. Let Fˆ = hx1 , . . . , xn , y1 , . . . , yn i be a free group of rank 2n. Then Fˆ /γ3 (Fˆ )Fˆ p is a free nilpotent group of class two and exponent p. Its derived subgroup is elementary abelian of order pn(2n−1) and, according to Lemma 2.6.3, it has a basis over Fp given by the set of basic commutators of weight 2 on x1 , . . . , xn , y1 , . . . , yn , namely {[xj , xk ], [yj , yk ] | 1 ≤ k < j ≤ n} ∪ {[yj , xk ] | j, k = 1, . . . , n}. Let us put R = h[xj , xk ], [yj , yk ] | 1 ≤ k < j ≤ ni and define F = Fˆ /γ3 (Fˆ )Fˆ p R. 2

Then F 0 is clearly elementary abelian of order pn and a basis of F 0 over Fp is given by the set {[yj , xk ] | j, k = 1, . . . , n}. The factor group F/F 0 is a free abelian group of exponent p and rank 2n; in other words, it is elementary abelian of order p2n . Let us fix an Fp -linear isomorphism τ : V ⊕ V → F/F 0 such that τ maps V ⊕0 and

onto

0⊕V

hx1 , . . . , xn iF 0 /F 0 ,

onto

hy1 , . . . , yn iF 0 /F 0 .

94


Let us make F/F 0 into an Fp Q-module isomorphic to V ⊕ V via τ , namely let us define an action of Q on F/F 0 according to the formula −1

(xF 0 )ξ = ((xF 0 )τ ξ)τ for all x ∈ F and for all ξ ∈ Q, and extend it Fp -linearly to an action of Fp Q on F/F 0 . Thus F/F 0 becomes an Fp Q-module and τ an Fp Q-module isomorphism. Furthermore, F/F 0 is the direct sum of the Fp Q-submodules hx1 , . . . , xn iF 0 /F 0 and hy1 , . . . , xn iF 0 /F 0 . Let us fix a generator ξ of Q. We shall show that the automorphism of F/F 0 induced by ξ can be lifted to an automorphism ξˆ of F . First of all, the automorphism induced by ξ on F/F 0 = Fˆ /Fˆ 0 Fˆ p can be lifted to an automorphism ξˆ of the free group Fˆ , and we may assume that the subgroups ˆ In Fˆ1 = hx1 , . . . , xn i and Fˆ2 = hy1 , . . . , yn i of Fˆ are left invariant by ξ. particular, the subgroups Fˆ10 = h[xj , xk ], γ3 (Fˆ1 ) | 1 ≤ k < j ≤ ni and

Fˆ20 = h[yj , yk ], γ3 (Fˆ2 ) | 1 ≤ k < j ≤ ni

ˆ or in other words, (Fˆ 0 )ξˆ = Fˆ 0 and (Fˆ 0 )ξˆ = Fˆ 0 . Since are left invariant by ξ, 1 1 2 2 ˆ γ3 (Fˆ ) and Fˆ p are characteristic subgroups of Fˆ , we also have that γ3 (Fˆ )ξ = ˆ γ3 (Fˆ ) and (Fˆ p )ξ = Fˆ p . As a consequence, the automorphism ξˆ of Fˆ induces ˆ We an automorphism of F = Fˆ /γ3 (Fˆ )Fˆ p Fˆ10 Fˆ20 , which we shall also call ξ. may assume that ξˆ has order |Q| (otherwise we may replace ξˆ with a suitable nt power ξˆp ), and thus we can regard F as a Q-group. In particular, F 0 can be regarded as a (semisimple) Fp Q-module. Now let us put E = hF 0 , x1 , . . . , xn i and let us apply Lemma 2.2.1 to the following subgroups of F : K3 = 1, H3 = K1 = F 0 , H1 = K2 = E, H2 = F. Thus we obtain an Fp -bilinear map δ : (E/F 0 ) × (F/E) → F 0 (xF 0 , yE) 7→ [x, y].


95

Since E is a Q-subgroup of F , the factor groups E/F 0 , F/E can also be regarded as Fp Q-modules. Thus we obtain an Fp Q-module homomorphism δ¯ : (E/F 0 ) ⊗ (F/E) → F 0 (xF 0 ) ⊗ (yE) 7→ [x, y]. Actually, δ¯ is an isomorphism. In fact, the set {(xj F 0 ) ⊗ (yk E) | j, k = 1, . . . , n} is a basis of (E/F 0 ) ⊗ (F/E) over Fp , and we have ¯

((xj F 0 ) ⊗ (yk E))δ = [xj , yk ] = [yk , xj ]−1 ; on the other hand, the elements [yk , xj ] of F 0 are distinct for j, k = 1, . . . , n and form a basis of F 0 over Fp , as we saw earlier. Thus δ¯ is an Fp Q-module isomorphism, and hence F 0 is isomorphic to V ⊗ V as an Fp Q-module. Now the Fp Q-module isomorphism τ : V ⊕ V → F/F 0 induces two Fp Q-module isomorphisms from V onto E/F 0 and F/E respectively, namely → E/F 0

τ1 : V

v 7→ (v, 0)τ , and τ2 : V

→ F/E

v 7→ (0, v)τ E. An Fp Q-module isomorphism τ1 ⊗ τ2 : V ⊗ V → (E/F 0 ) ⊗ (F/E) is induced in an obvious way, and we have the following commutative diagram of Fp Q-module homomorphisms: δ¯

(E/F 0 ) ⊗ (F/E) −−−→ F 0 x  τ1 ⊗τ2 

V ⊗V

  δ¯−1 (τ1 ⊗τ2 )−1 γ¯ y

γ ¯

−−−→ V

96 Let us put


¯

K = ker (δ¯−1 (τ1 ⊗ τ2 )−1 γ¯ ) = (ker γ¯ )(τ1 ⊗τ2 )δ .

The factor group F 0 /K is then an Fp Q-module and the Fp Q-module epimorphism δ¯−1 (τ1 ⊗ τ2 )−1 γ¯ : F 0 → V induces an Fp Q-module isomorphism ν : F 0 /K → V such that δ¯−1 (τ1 ⊗ τ2 )−1 γ¯ = πν, where π : F 0 → F 0 /K is the natural epimorphism. Since K is also a central subgroup of F , we can form the factor group X = F/K, which is a p-group of class 2, exponent p and order p3n . Let us put L = E/K. Then L is an elementary abelian normal subgroup of X and has order p2n . ˆ having order |Q|) Let ξ¯ denote the automorphism of X (like ξ and ξ, ¯ be the subgroup of Aut(X) induced by the automorphism ξˆ of F and let Q ¯ ¯ ¯ generated by ξ. Let σ : Q → Q be the group isomorphism such that ξ σ = ξ. ¯ ¯ Then the Q-group X becomes a Q-group of Q-length 3 via σ −1 . The series 1 < X0 < L < X ¯ ¯ is a Q-composition series of X and the Q-composition factors of X, regarded as Fp Q-modules via σ, are all isomorphic to V . Furthermore, the factor group X/X 0 is elementary abelian, and regarded as an Fp Q-module it is isomorphic to V ⊕ V . In particular, X 0 is the Frattini subgroup of X. ¯ Furthermore, we have that Z(X) = X 0 . In fact, Z(X) is a Q-subgroup 0 0 of X, and it contains X . Let us suppose for a moment that X < Z(X). It ¯ follows that Z(X) has Q-length 2, because Z(X) cannot be the whole of X, which is not abelian. If Z(X) contained L, then from the fact that [L, X] = 1, it would follow that [E, F ] ≤ K; this would contradict the fact that the map δ¯ defined above is an isomorphism. We deduce that Z(X) 6= L, and thus that ¯ Z(X)L = X, because X/X 0 has Q-length 2. It follows that X 0 = L0 = 1, which contradicts the fact that X is not abelian. As a consequence, our assumption is wrong, and therefore we get that Z(X) = X 0 . For each ϕ¯ ∈ HomFp Q¯ (X/X 0 , X 0 ),


97

the map w ¯ϕ¯ defined by xw¯ϕ¯ = x(xX 0 )ϕ¯ for all x ∈ X, ¯ The set is an automorphism of X commuting with ξ. ¯ = {w¯ϕ¯ | ϕ¯ ∈ HomF Q¯ (X/X 0 , X 0 )} W p ¯ equals the order of X/X 0 , is a subgroup of AutQ¯ (X), and the order of W ¯ then D2 satisfies namely p2n . In fact, if we put D2 = X, N2 = X 0 and Q2 = Q, ¯ is exactly the subgroup W2 of conditions (1), . . . , (6) of Section 5.2, and W AutQ2 (D2 ) which is defined there. Let us define G2 = [D2 ](W2 × Q2 ) as in Section 5.2. The group G2 will also be called H here and we will therefore write ¯ × Q). ¯ H = [X](W Since all assumptions of Section 5.2 are satisfied, Lemma 5.2.1 applies and yields that G and H have identical character tables, and that H 0 = X, whence H 00 = X 0 and H 000 = 1. Thus G and H have identical character tables and G is metabelian, as we said earlier, while H has derived length 3. ¯ Let us consider the following Q-composition series of X, which is also part of a chief series of H going from 1 to X: 1 < X 0 < L < X. According to the analysis of commutation in X which we carried out in ¯ Section 3.4, the Fp Q-module epimorphism associated with our chief series is the following: ¯ : (L/X 0 ) ⊗ (X/L) → X 0 δπ (xX 0 ) ⊗ (yL) 7→ [x, y], where the factor groups L/X 0 and X/L have been identified with E/F 0 and F/E respectively. If we regard the Fp Q-modules L/X 0 , X/L and X 0 as ¯ is also an Fp Q¯ Fp Q-modules via σ, then the Fp Q-module epimorphism δπ ¯ coincides with our prescribed Fp Q-module module epimorphism. As such, δπ

98


epimorphism γ¯ : V ⊗ V → V , after identifying L/X 0 , X/L and X 0 with V by means of suitable Fp Q-module isomorphisms. In other words, the following diagram is commutative: ¯ δπ

(L/X 0 ) ⊗ (X/L) −−−→ X 0 x  τ1 ⊗τ2 

V ⊗V

  ν y

γ ¯

−−−→ V.

As we promised, in analogy with what we did in Section 5.3 for the map γ¯ , under the assumption p 6= 2 we have constructed pairs of groups (G, H) which satisfy Hypotheses 3.1.2 and which give rise,through the method of Section 5.2, to any arbitrarily given Fp Q-module epimorphism γ¯ : V ⊗ V → V . It is perhaps worth remarking that 1 < X0 < L < X ¯ is not in this case the unique Q-composition series of X, nor the unique chief ¯ series of H passing through 1 and H. If we choose a different Q-composition series ¯ < X, 1 < X0 < L ¯ is not abelian. In that case, our analysis of commutait may happen that L tion in X would produce a non-trivial skew-symmetric Fp -bilinear map ¯ 0 ) × (L/X ¯ 0) → X 0, δ 0 : (L/X and thus we would fall again into the unary case, for which we have already built examples in Section 5.3. However, the situation described above cannot happen if our Fp Q-module V is a composition factor of V ⊗ V , but not of V ∧ V (for example, if V has dimension 2 or 3 over Fp ); this shows that the binary case gives rise to examples which are genuinely different from those of the unary case. Now, we observe that the normal Sylow p-subgroups of G and H, namely ¯ , have identical character tables but are not isomorP = AW and P¯ = X W phic. We omit the proof of this fact, which consists of a counting argument similar to that which we used in the proof of the corresponding fact of Section 5.3. We only observe that the argument here is slightly more complicated,

5.6. MATRIX REPRESENTATIONS FOR THE BINARY CASE

99

because A/P 0 is not a chief factor of G; therefore, in addition to the set AP of abelian subgroups of P which have order p3n and contain P 0 , one needs to consider also the set BP of unordered pairs {B, C} of elements of AP such that BC = P (and similar sets AP¯ and BP¯ concerning P¯ ). We conclude this section by computing the smallest possible order of G and H for a pair (G, H) of groups constructed as above. It follows easily from the case-study which concludes Section 3.5 that if p is an odd prime, Q is a non-trivial cyclic p0 -group, and V is faithful irreducible module for Q over Fp such that V ⊗ V has a composition factor isomorphic to V , then the smallest possible order of V is attained when V has dimension 2 over Fp , the group Q has order 3, and the prime p is 5. In that case, V has 25 elements, and the order of G is as small as possible, namely |G| = 510 · 3; however, this is bigger that 312 · 5, that is the order of the smallest groups G and H which we constructed in Section 5.3.

5.6

Matrix representations for the binary case

In this section we shall construct explicit matrix representations for some of the groups of Section 5.5. We shall represent faithfully G as a subgroup of GL4 (pn ), and H as a subgroup of GL5 (pn ). Let us fix an odd prime p and a positive integer q such that pi + 1 ≡ pj mod |Q| for some integers i, j (and we may always assume that 0 ≤ i ≤ n/2

and

0 ≤ j < n).

Let Q = hξi be a cyclic group of order q, and let ε be a primitive q-th root of unity in Fpn . Let Vε be a cyclic group of order q, and let ε be a primitive qth root of unity in Fpn . Let Vε be the faithful irreducible module for Q over Fp which is defined in Theorem 2.3.1. We shall also consider the Fp Q-modules Vεpi and Vεpj defined similarly, which are both isomorphic to Vε . Because of our assumptions on p and q, the module Vε appears as a

100


composition factor of the tensor square module Vε ⊗ Vε . We shall explicitly define an Fp Q-epimorphism from Vε ⊗ Vεpi onto Vεpj . For all pure tensors x ⊗ y of Vε ⊗ Vεpi , let us define (x ⊗ y)γ¯ = xy. Since the expression xy is Fp -bilinear in (x, y), it follows that γ¯ extends to an Fp -linear map γ¯ : Vε ⊗ Vεpi → Vεpj . Furthermore, the map γ¯ is an Fp Q-module homomorphism, because we have i

((x ⊗ y)ξ)γ¯ = ((εx) ⊗ (εp y))γ¯ i

= ε1+p xy i

= ε1+p (x ⊗ y)γ¯ j

= εp (x ⊗ y)γ¯ = (x ⊗ y)γ¯ ξ. It is clear that γ¯ is not the zero homomorphism. Thus γ¯ is an Fp Q-module epimorphism, because Vεpj is an irreducible Fp Q-module. (It should be said that not all Fp Q-module epimorphisms from V ⊗ V onto V can be put into this particularly simple form by suitably identifying V with Vε , Vεpi and Vεpj .) With this particular choice of the Fp Q-epimorphism γ¯ , let us define the group X = F/K as we did in Section 5.5. Let X3 be the set of the matrices 



1 a c  1 b   , 1 with a, b, c ∈ Fpn ; hence X3 is a Sylow p-subgroup of GL3 (pn ). Moreover, X3 is isomorphic to X, and it is not too difficult to construct an epimorphism from F onto X, with kernel K. It is also clear that the diagonal matrix   ξ¯3 = 



1

 

ε 1+pi

ε

normalizes X3 , and induces by conjugation an automorphism of X3 which corresponds to the automorphism ξ¯ of X defined in Section 5.5 (for a suitable

5.6. MATRIX REPRESENTATIONS FOR THE BINARY CASE

101

¯ of H choice of the isomorphism X → X3 ). Thus the normal subgroup X Q n is isomorphic to a subgroup of GL3 (p ). Now we are ready to embed the whole of H into GL5 (pn ). First of all, we observe that the set X5 of the matrices 

j

1 a ap   1   1 

bp

 



j

c  b      

1 1

with a, b, c ∈ Fpn is a subgroup of GL5 (pn ) isomorphic to X3 (and hence to X). In fact, an explicit isomorphism from X3 onto X5 is given by the map 

1 a ap    1 a c  1   pi  7→   1 b  1   1 

j



j

bp

c i  bp  

.   

1 1

The diagonal matrix     ¯ ξ5 =    



1

      

ε j

εp

j

εp

εp

j

normalizes X5 . The automorphism of X5 induced by ξ¯5 by conjugation corresponds to the automorphism of X3 induced by ξ¯3 by conjugation (with respect to the given isomorphism from X3 onto X5 ) and thus to the automorphism ξ¯ of X (with respect to a suitable isomorphism from X onto X5 ). ¯ 5 of GL5 (pn ) consisting of the matrices Now the subgroup W        



1 1 1 1

   d   e  

1

102


with d, e ∈ Fpn is elementary abelian of order p2n , centralizes ξ¯5 , and nor¯ × Q) ¯ is isomorphic to the malizes X5 . It is quite clear now that H = [X](W ¯ 5 hξ¯5 i of GL5 (pn ), namely the group of the matrices subgroup H5 = X5 W 

j

1 a ap   εl   lpj  ε   

j

bp

εlp

j

c i bp d e lpj ε

     ,   

with a, b, c, d, e ∈ Fpn and l = 1, . . . , |Q|. Thus we have constructed a faithful matrix representation of H. It is much easier to give a matrix representation of the group G. In fact, it is easy to see that G is isomorphic to the subgroup G4 of GL4 (pn ) which consists of the matrices   1 a b c  εl d       εl e  εl with a, b, c ∈ Fpn and l = 1, . . . , |Q|. Let us also notice that G is isomorphic to the subgroup G5 of GL5 (pn ) which consists of the matrices 

j

1 a ap  εl   j  εlp   

ε

b

c

lpj

d e j εlp

       

with a, b, c ∈ Fpn and l = 1, . . . , |Q|. In fact an isomorphism from G3 onto G5 is given by the map 

j

j

1 a ap bp 1 a b c  εl    εl d  j   εlp   7→    εl e  j   εlp εl 



j

cp

j

dp j ep j εlp

    .   

5.7. POWER-MAPS

103

Since G5 and H5 normalize each other, G5 H5 is a subgroup of GL5 (pn ). Clearly G5 H5 consists of the matrices 

j

1 a ap  εl   j  εlp   

b

εlp

j

c f d e lpj ε

    ,   

with a, b, c, d, e, f ∈ Fpn and l = 1, . . . , |Q|. The groups G5 and H5 are normal subgroups of G5 H5 of index pn . In particular, we have obtained that our groups G and H can be simultaneously embedded as normal subgroups into a group of order pn |G|. We finally observe that our assumption that the prime p is odd is unnecessary for the construction of our groups of matrices (like it was in Section 5.4). Of course, when p = 2, the normal Sylow 2-subgroups of G and H will have exponent 4 = p2 instead of p. If we allow p = 2, then the smallest possible order for G5 and H5 becomes 210 · 3, which is smaller than the minimal order of the matrix groups G4 and H4 of Section 5.4. Indeed, this is the smallest example of a pair of groups satisfying Hypotheses 3.1.2 which we were able to construct.

5.7

Power-maps

In this last section we shall show that for most of the pairs of groups (G, H) which we have constructed in this chapter, G and H have not only identical character tables, but also identical character tables with power-maps. If K is a conjugacy class of G and m is an integer, then there is a conjugacy class K[m] of G which consists of the mth powers of the elements of K. The maps K 7→ K[m] from the set of the conjugacy classes of G into itself are usually called power-maps. When the power-maps are added to the character table of a group G (in some way which we shall not formalize here), the resulting object gives considerably more information about G than the character table alone does. In particular, the power-maps determine the order of the elements of any given conjugacy class, and thus sometimes allow one to distinguish between groups which have identical character tables, like for instance D8 and Q8 (or, more generally, the two non-isomorphic extraspecial

104


p-groups of a given order). We should say, though, that the prime factors of the orders of the elements are determined by the character table alone, as shown in [13, Theorem (8.21)]. If two groups G and H have identical character tables, via some bijections α and β, the additional condition that G and H also have identical powermaps can be expressed by requiring that (Kα )[m] = (K[m] )α for all conjugacy classes K of G and for all integers m. We shall give instead the following equivalent definition. Definition 5.7.1 Let G1 and G2 be finite groups. We shall say that G1 and G2 have identical character tables with power-maps if there exist bijections α : G1 → G2 and β : Irr(G1 ) → Irr(G2 ), such that for all integers m we have χβ ((g α )m ) = χ(g m ) for all g ∈ G1 and for all χ ∈ Irr(G1 ). We observe that when m is composite, the mth power-map K 7→ K[m] of a group G is completely determined by the set of the pth power-maps with p ranging over the prime divisors of m. Furthermore, we can restrict our attention to the primes p which divide the order of G. In fact, the mth power-maps for (m, |G|) = 1 (which, incidentally, are the only power-maps which are bijective) are uniquely determined by the character table of G, as the following well-known lemma shows. Lemma 5.7.2 Let G1 and G2 have identical character tables via the bijections α : G1 → G2 , β : Irr(G1 ) → Irr(G2 ), and let m be an integer with (m, |G|) = 1. Then we have χβ ((g α )m ) = χ(g m ) for all g ∈ G1 and for all χ ∈ Irr(G1 ).

5.7. POWER-MAPS

105

Proof Let n = |G1 |, and let E be the splitting field for the polynomial xn − 1 over Q in C. Then E = Q[ε], where ε is a primitive nth root of 1 in C. From the fact that (m, |G|) = 1 it follows that εm is also a primitive nth root of 1. Now let G be the Galois group of E over Q. Since the cyclotomic polynomials over Q are irreducible (see for instance [15, Theorem 4.17]), the primitive nth roots of 1 in C are transitively permuted by G. Hence there exists σ ∈ G such that εσ = εm . Now let χ ∈ Irr(G1 ) and g ∈ G1 . If X is a complex representation of G1 affording the character χ, then, according to [13, Lemma (2.15)], X(g) is similar to a diagonal matrix diag(εi1 , . . . , εif ), where f = χ(1) and i1 , . . . , if are integers. In particular, χ(g) =

f X

εij .

j=1

It follows that X(g m ) = X(g)m is similar to diag(εi1 m , . . . , εif m ), and hence χ(g m ) =

f X

εij m =

f X



(εσ )ij = 

j=1

j=1

f X

σ

εij  = χ(g)σ .

j=1

In a similar way we obtain that χβ ((g α )m ) = χβ (g α )σ . Since χβ (g α ) = χ(g), the conclusion now follows.

¤

A straightforward consequence of Lemma 5.7.2 is the following: if two pgroups of exponent p have identical character tables, then they have identical character tables with power-maps. This fact was employed by Dade in [6], where he gave the first examples of non-isomorphic groups having identical character tables with power-maps, as an answer to a question of Brauer [19, Problem 4]. Let us notice in passing that Dade’s proof that his p-groups have identical character tables was a special case of our Theorem 4.3.1. Now, our Corollary 4.3.2 can be easily adapted in order to handle character tables with power-maps. Let us see only a special case.

106


Theorem 5.7.3 Let Ni be an abelian normal subgroup of Gi , for i = 1, 2. Let us suppose that following condition holds, in addition to hypotheses (i), (ii), (iii) of Corollary 4.3.2: (iv) hgi ∩ Ni = 1 for all g ∈ Gi \ Ni , for i = 1, 2. Then G1 and G2 have identical character tables with power-maps. Proof Let us construct the maps α and β according to the proof of Corollary 4.3.2. Then we have χβ (g α ) = χ(g) for all g ∈ G1 and for all χ ∈ Irr(G1 ). Let us fix an integer m. Although it is not necessarily true that (g α )m = (g m )α for all g ∈ G1 (which would conclude the proof), we do have that (g α˜ )m = (g m )α˜ for all g ∈ N1 , and that ((gN1 )α¯ )m = (g m N1 )α¯ for all g ∈ G1 , because α ˜ and α ¯ are group isomorphisms. As a first consequence, the assertion χβ ((g α )m ) = χ(g m ) for all χ ∈ Irr(G1 ), is certainly true for g ∈ N1 . Furthermore, it is also true for those g ∈ G1 \ N1 such that g m ∈ G1 \ N1 . In fact, this implies that (g α )m ∈ G2 \ N2 , because α ¯ is a group isomorphism; hence χβ ((g α )m ) = 0 = χ(g m ) for all χ ∈ Irr(G1 ) \ Irr(G1 /N1 ). On the other hand, if χ ∈ Irr(G1 /N1 ) we have ¯

χβ ((g α )m ) = χβ ((g α )m N2 ) ¯

= χβ (((gN1 )α¯ )m ) ¯

= χβ ((g m N1 )α¯ ) = χ(g m N1 ) = χ(g m ).

5.7. POWER-MAPS

107

Now we are left with the case of an element g of G1 \N1 such that g m ∈ N1 . In this case we have (g α )m ∈ N2 , again because α ¯ is a group isomorphism. m According to hypothesis (iv), we have g = 1 and (g α )m = 1. Consequently, we have χβ ((g α )m ) = χβ (1) = χ(1) = χ(g m ). This concludes the proof.

¤

Now let us assume the hypotheses (1), . . . , (6) of Section 5.2, and let us assume in addition that the groups D1 and D2 have exponent p; in particular, the prime p must be odd. Let us define the groups Gi = [Di ](Wi ⊗ Qi ), for i = 1, 2, as we did in Section 5.2. Then the normal Sylow p-subgroup Di Wi of Gi has exponent p (for i = 1, 2); in fact, since Di Wi has class two and both of Di and Wi have exponent p, we have p (xw)p = xp wp [w, x](2) = 1

for all x ∈ Di and w ∈ Wi , according to [10, Kapitel III, Hilfssatz 1.3 b)]. As we proved in Lemma 5.2.1, G1 and G2 have identical character tables. Now Theorem 5.7.3 allows us to prove that G1 and G2 have identical character tables with power-maps. Indeed, let us define isomorphisms α ¯ : G1 /N1 → G2 /N2 and α ˜ : N1 → N 2 as in the proof of Lemma 5.2.1; as we proved there, hypotheses (i), (ii), and (iii) of Lemma 4.3.2 (and thus of Theorem 5.7.3) are satisfied. Hence it remains to check hypothesis (iv) of Theorem 5.7.3, namely that hgi ∩ Ni = 1 for all g ∈ Gi \ Ni , for i = 1, 2. This is clearly true for g ∈ Di Wi \ Ni , because Di Wi has exponent p. Let g ∈ Gi \ Di Wi . Then the order of g is not a power of p, and we can choose a prime divisor q of the order |g| of g, distinct from p. Hence h = g |g|/q has order q; consequently, h belongs to some Sylow q-subgroup of Gi . On

108


the other hand, Qi contains a Sylow q-subgroup of Gi . It follows that h is conjugate to some (non-identity) element of Qi . Now, every element of Qi different from the identity element acts fixedpoint-freely on Di by conjugation. Consequently, h acts fixed-pont-freely on Di by conjugation; in other words, CG (h) ∩ Di = 1. Because hgi ≤ CG (h) and Ni ≤ Di , we have that hgi ∩ Ni = 1. Thus hypothesis (iv) of Theorem 5.7.3 is also satisfied. We conclude that the groups G1 and G2 have identical character tables with power-maps, as claimed. In particular, for all the examples (G, H) which we constructed in Sections 5.3 and 5.5, which satisfy the assumption p 6= 2, we obtain that G and H have identical character tables with power-maps.

Chapter 6 Nilpotent counterexamples In Chapter 4 we had a fairly deep insight into the structure of a minimal counterexample (G, H) to Conjecture 3.1.1. In fact, the results of Chapter 4 strongly suggest that the basic pattern for the construction of G and H is essentially that of our examples of Chapter 5. However, behind our investigations of Chapter 4 there was a fundamental assumption, namely that G and H were not nilpotent (Hypotheses 3.1.2). In this chapter we shall turn our attention to p-groups. Although our knowledge about nilpotent counterexamples to Conjecture 3.1.1 is very limited, we shall be able to construct such a counterexample. Let us briefly sketch how this example originates. We aim to construct p-groups G1 and G2 , such that G001 = 1 and G002 6= 1, and G1 , G2 have identical character tables. Since the character table of a nilpotent group determines its nilpotency class, G1 and G2 must have the same class c; necessarily c is at least 4, because γ4 (G2 ) ≥ G002 6= 1. Thus, the smallest example we can hope for will have |γ4 (G1 )| = |γ4 (G2 )| = p and G001 = 1, G002 = γ4 (G2 ). Our basic tool for comparing character tables is Corollary 4.3.2. In order to be able to apply that corollary with Ni = γ4 (Gi ) for i = 1, 2, we shall require that G1 /γ4 (G1 ) ∼ = G2 /γ4 (G2 ). But then we may as well regard G1 and G2 as factor groups of the same group G; for example, we may take G to be the direct product of G1 and G2 with amalgamated factor groups G1 /γ4 (G1 ) ∼ = G1 /γ4 (G2 ) (see [10, Kapitel I, Satz 9.11]). 109

110

CHAPTER 6. NILPOTENT COUNTEREXAMPLES

In the last analysis, we are looking for a p-group G of class 4, in which γ4 (G) is the direct product of two cyclic groups Z1 , Z2 of order p with Z1 = G00 , and such that (G/Z1 , γ4 (G)/Z1 ) and (G/Z2 , γ4 (G)/Z2 ) are both Camina pairs (which is hypothesis (iii) of Corollary 4.3.2). Let us put Gi = G/Zi and Ni = γ4 (G)/Zi , for i = 1, 2. According to Lemma 4.2.1, the condition that (G1 , N1 ) and (G2 , N2 ) are Camina pairs is equivalent to Ni ⊆ bg, Gi c for all g ∈ Gi \ Ni , for i = 1, 2. This condition can be easily reformulated in terms of G, as follows: (bg, Gc ∩ γ4 (G)) · Zi = γ4 (G) for all g ∈ G \ γ4 (G), for i = 1, 2. We only observe that because γ4 (G) is a central subgroup of G, the set bg, Gc ∩ γ4 (G) is a subgroup of γ4 (G) (though it may be strictly contained in [g, G] ∩ γ4 (G)); this fact can easily be proved directly, or it can be viewed as an application of Lemma 2.2.4. It will be useful to keep in mind the above condition during the course of the construction. Now let us proceed with the details of the construction. Let F be the free group on three generators x, y and z. Let us fix a prime p ≥ 5, and define F¯ = F/γ5 (F )F p , where F p denotes the (fully invariant) subgroup of F generated by the pth powers of all elements of F . Then F¯ is a nilpotent group of class 4 and exponent p (actually F¯ is the 3-generator free object in the variety of nilpotent groups of class 4 and exponent p). We know from Theorem 2.6.3 that the factor groups γn (F¯ )/γn+1 (F¯ ) for n = 1, 2, 3, 4 are elementary abelian, and that a set of representatives for a basis of γn (F¯ )/γn+1 (F¯ ) as a vector space over the field Fp is given by the set Cn of basic commutators of weight n. Explicitly, we have: C1 = {x, y, z}, C2 = {[y, x], [z, x], [z, y]}, C3 = {[y, x, x], [z, x, x], [y, x, y], [z, x, y], [z, y, y], [y, x, z], [z, x, z], [z, y, z]},

111 C4 = {[y, x, x, x], [z, x, x, x], [y, x, x, y], [z, x, x, y], [y, x, y, y], [z, x, y, y], [z, y, y, y], [y, x, x, z], [z, x, x, z], [y, x, y, z], [z, x, y, z], [z, y, y, z], [y, x, z, z], [z, x, z, z], [z, y, z, z], [[z, x], [y, x]], [[z, y], [y, x]], [[z, y], [z, x]]}. In the next lemma we shall define a certain factor group H of F¯ , which is a ‘first approximation’ of the group G that we are looking for. We recall that in a group of class 4, like F¯ , we have the identity [h, g, k] = [g, h, k]−[h,g] = [g, h, k]−1 ; furthermore, the Witt’s identity [g, h, k g ] [k, g, hk ] [h, k, g h ] = 1 assumes the following form: [g, h, k] [[g, h], [k, g]] [k, g, h] [[k, g], [h, k]] [h, k, g] [[h, k], [g, h]] = 1. Lemma 6.0.4 For any prime p ≥ 5, there exists a group H = hx, y, zi of order p10 , exponent p and class 4, such that the following conditions are satisfied: (i) the subset C¯n of H defined below, for i = 1, 2, 3, 4, forms a set of representatives of a basis of γn (H)/γn+1 (H), viewed as a vector space over Fp , the field of p elements: C¯1 = {x, y, z}, C¯2 = {[y, x], [z, x]}, C¯3 = {[y, x, x], [z, x, x], [y, x, z]}, C¯4 = {[y, x, x, z], [[z, x], [y, x]]}; (ii) the following relations hold [z, x, x, y] = [y, x, x, z] [[z, x], [y, x]]−2 , [z, x, y] = [y, x, z] [[z, x], [y, x]]−1 ;

112


(iii) in addition, the relation c = 1 holds for each basic commutator c not appearing in the diagram in (i) or in the relations in (ii). Proof We shall proceed in three steps. Step 1. Let us define the following subgroup of F¯ : R4 = hC4 \ {[z, x, x, y], [y, x, x, z], [[z, x], [y, x]]}, [z, x, x, y]−1 [y, x, x, z] [[z, x], [y, x]]−2 i. Since R4 ≤ γ4 (F¯ ) ≤ Z(F¯ ) (actually, it is not too difficult to prove that γ4 (F¯ ) = Z(F¯ )), we have that C1 , C2 , C3 , and C¯4 are sets of representatives of bases of γn (F¯ /R4 )/γn+1 (F¯ /R4 ) for n = 1, 2, 3, 4 respectively. Step 2. Let us define the following subgroup of F¯ : R3 = hR4 , [y, x, y], [z, y, y], [z, x, z], [z, y, z], [z, x, y]−1 [y, x, z] [[z, x], [y, x]]−1 i. Then R3 /R4 ≤ Z(F¯ /R4 ). We shall sketch a proof of this fact. First of all, the basic commutators [y, x, y], [z, y, y], [z, x, z], [z, y, z] are central in F¯ /R4 . In fact, the commutator of any of these elements with x, y or z (for instance [[z, y, y], x] = [z, y, y, x]) is a (not necessarily basic) commutator of weight 4 in the letters x, y, z, in which either y or z appears at least twice; by a repeated application of the Witt’s identity, this commutator can be written as a product of basic commutators of weight 4 and their inverses (we recall that since we are working in a group of class 4, all commutators of higher weight are trivial) in which, again, either y or z appears at least twice; such basic commutators all belong to R4 , and what we claimed is proved. Since the computations which we outlined are easy, but tedious, let us see just one example: [z, y, y, x] = [[z, y], [y, x]] [z, y, x, y] = [[z, y], [y, x]] [y, x, z, y]−1 [z, x, y, y] = [[z, y], [y, x]] ([y, x, y, z]−1 [[y, x], [z, y]]−1 ) [z, x, y, y] = [[z, y], [y, x]]2 [y, x, y, z]−1 [z, x, y, y] ∈ R4 . Now, we observe that [[z, x], [y, x]] is central in F¯ /R4 ; furthermore, the product [z, x, y]−1 [y, x, z] is also central in F¯ /R4 , because the elements y and

113 z centralize both of [z, x, y] and [y, x, z] (mod R4 ), while we have [[z, x, y]−1 [y, x, z], x] = [z, x, y, x]−1 [y, x, z, x] = [z, x, x, y]−1 [[z, x], [y, x]]−1 [y, x, x, z] [[y, x], [z, x]] = [z, x, x, y]−1 [y, x, x, z] [[z, x], [y, x]]−2 ∈ R4 . Hence [z, x, y]−1 [y, x, z] [[z, x], [y, x]]−1 is also central in F¯ /R4 . We conclude that R3 /R4 ≤ γ3 (F¯ /R4 ) ∩ Z(F¯ /R4 ). Since we also have R3 /R4 ∩ γ4 (F¯ /R4 ) = 1, it follows that sets of representatives of bases of γn (F¯ /R3 )/γn+1 (F¯ /R3 ) for n = 1, 2, 3, 4 are given by C1 , C2 , C¯3 , C¯4 respectively. Step 3. Let us define the following subgroup of F¯ : R2 = hR3 , [z, y]i. Then R2 /R3 ≤ Z(F¯ /R3 ). In fact, [z, y] clearly commutes with y and z (mod R3 ); it commutes with x too, (mod R3 ), because the Witt’s identity [z, y, xz ] [x, z, y x ] [y, x, z y ] = 1 read (mod R3 ) becomes [z, y, x] [x, z, y] [[x, z], [y, x]] [y, x, z] ≡ 1 mod R3 , and thus we get that [z, y, x] ≡ [y, x, z]−1 [[x, z], [y, x]]−1 [x, z, y]−1 = [[z, x], [y, x]] [y, x, z]−1 [z, x, y] = ([z, x, y]−1 [y, x, z] [[z, x], [y, x]]−1 )−1 ≡ 1 mod R3 . Hence R2 /R3 ≤ γ2 (F¯ /R3 ) ∩ Z(F¯ /R3 ). This fact, together with the fact that R2 /R3 ∩ γ3 (F¯ /R3 ) = 1, implies that sets of representatives of bases of γn (F¯ /R2 )/γn+1 (F¯ /R2 ) for n = 1, 2, 3, 4 are given by C1 = C¯1 , C¯2 , C¯3 , C¯4 respectively. Let us define H = F¯ /R2 . Then H has exponent p and clearly satisfies the conditions (i), (ii) and (iii). In particular H has order p10 and class 4. This completes the proof of the lemma. ¤

114


Lemma 6.0.4 implicitly gives a (rather long) presentation of H in terms of generators (namely x, y and z) and relations. It would be easy to prove that a shorter presentation is H = hx, y, z | xp = y p = z p = 1, [z, y] = [y, x, y] = [z, x, z] = [y, x, x, x] = [z, x, x, x] = 1, [g1 , g2 , g3 , g4 , g5 ] = 1 for all g1 , . . . , g5 ∈ Hi. We said earlier that the group H should be a ‘first approximation’ of the group G that we are trying to construct. In fact, it would be possible to show that bg, Hc ∩ γ4 (H) 6= 1 for all g ∈ H \ γ4 (H). However, H does not satisfy the stronger condition which we required, namely (bg, Hc ∩ γ4 (H)) · H 00 = γ4 (H) for all g ∈ H \ γ4 (H), because we have for instance b[y, x], Hc ∩ γ4 (H) = h[[z, x], [y, x]]i = H 00 . We shall quickly remedy this problem. The assignments  u   x  

yu zu

= x[z, x, x]−1 = y = z

 v   x  

yv zv

= x[y, x, x]−1 = y = z

clearly define two mutually commuting automorphisms u, v of the group F¯ . Actually, they also define automorphisms u, v of its factor group H = ¯ F /R2 . In fact, u and v fix each element of γ3 (F¯ ), and of course they fix [z, y]. Since R2 = (R2 ∩ γ3 (F¯ )) · h[z, y]i, it follows that u, v centralize R2 . Hence they lift to automorphisms u, v of H = F¯ /R2 . Since the automorphisms u and v of H have order p (because they also have order p as automorphisms of F¯ ) the group K = hu, vi is an elementary abelian group of automorphisms of H, of order p2 . Let us consider the semidirect product G = [H]K. It has order p12 and class 4. Since we assumed that p ≥ 5, the p-group G is regular (see for example [10, Kapitel III, Satz 10.2 a)]); therefore G has exponent p, because H and K have exponent p.

115 The following equalities which hold in G will be useful in computations: [y, x, u] = [z, x, x, y], [z, x, u] = 1,

[y, x, v] = 1, [z, x, v] = [y, x, x, z].

Let us define two central subgroups of G, namely Z1 = h[[z, x], [y, x]]i = G00 , Z2 = h[y, x, x, z][[z, x], [y, x]]s i, where s is an integer such that s 6≡ 0, −1, −2 (mod p) (for instance s = 1). The factor groups G1 = G/Z1 and G2 = G/Z2 are groups of order p11 . Now we state our final result. Theorem 6.0.5 The groups G1 and G2 have identical character tables, but G1 is metabelian, while G2 has derived length three. Proof It is clear that the derived lengths of G1 and G2 are 2 and 3 respectively. In order to prove that G1 and G2 have identical character tables, we shall appeal to Corollary 4.3.2, with N1 = γ4 (G1 ) = γ4 (G)/Z1 , N2 = γ4 (G2 ) = γ4 (G)/Z2 , α ¯ : G/γ4 (G) → G/γ4 (G) the identity map and α ˜ : N1 → N 2 any group isomorphism, which is necessarily a G/γ4 (G)-module isomorphism, the central subgroups N1 of G1 and N2 of G2 beeing regarded as trivial G/γ4 (G)-modules by conjugation. Then the conditions (i) and (ii) of the corollary are satisfied. It remains to verify condition (iii), namely that (G1 , N1 ) and (G2 , N2 ) are Camina pairs, or equivalently that Ni ⊆ bg, Gi c for all g ∈ Gi \ Ni , for i = 1, 2.

116


We shall distinguish two cases. Case 1: Either g ∈ γ3 (Gi ) \ γ4 (Gi ) or g ∈ Gi \ hγ2 (Gi ), u, vi. Let us regard the elementary abelian groups V1 = γ3 (G)/γ4 (G) and V2 = G/hγ2 (G), u, vi as vector spaces of dimension 3 over Fp . Then the ordered sets {[y, x, x], [z, x, x], [y, x, z]} and {x, y, z} are sets of representatives of bases of V1 and V2 respectively. Let us fix also the bases {[y, x, x, z]} of γ4 (G)/Z1 and {[[z, x], [y, x]]} of γ4 (G)/Z2 . We have [γ3 (G), G] ⊆ γ4 (G), [γ3 (G), hγ2 (G), u, vi] = 1 and [γ4 (G), G] = 1; according to Lemma 2.2.1, commutation in G gives rise in a natural way to a Z-bilinear map (actually Fp -bilinear, because V1 , V2 are vector spaces over Fp ) γ : V1 × V2 → γ4 (G). Let πi : γ4 (G) → γ4 (G)/Zi be the natural homomorphisms for i = 1, 2. We shall show that the composite maps γπi : V1 × V2 → γ4 (G)/Zi are non-degenerate. We compute: ¯

¯

¯

[[y, x, x]a [z, x, x]b [y, x, z]c , xa¯ y b z c¯] = [y, x, x, z]a¯c+bb+c¯a [[z, x], [y, x]]−2bb−c¯a . Thus the map γπ1 has matrix 

1

 

1

  

1 with respect to the given bases of V1 , V2 and γ4 (G)/Z1 , and hence γπ1 is non-degenerate. On the other hand, the map γπ2 has matrix 

−s

 

−2 − s −1 − s

  

117 with respect to the given bases of V1 , V2 and γ4 (G)/Z2 ; hence γπ2 is nondegenerate, because s 6≡ 0, −1, −2 (mod p). The non-degeneracy of γπ1 and γπ2 means that [g, Gi ] ⊇ Ni , for all g ∈ γ3 (Gi ) \ γ4 (Gi ), for i = 1, 2, and that [γ3 (Gi ), g] ⊇ Ni , for all g ∈ Gi \ hγ2 (Gi ), u, vi, for i = 1, 2. It follows in particular that [g, Gi ] ⊇ Ni for all g ∈ (γ3 (Gi ) \ γ4 (Gi )) ∪ (Gi \ hγ2 (Gi ), u, vi), for i = 1, 2. Case 2: g ∈ hγ2 (Gi ), u, vi \ γ3 (Gi ). Let us regard the elementary abelian group W = g ∈ hγ2 (G), u, vi/γ3 (G) as a vector space of dimension 4 over Fp . The ordered set {u, v, [y, x], [z, x]} is then a set of representatives of a basis of W . We have hγ2 (G), u, vi0 ≤ γ4 (G), and [hγ2 (G), u, vi, γ3 (G)] = 1; according to Lemma 2.2.1, commutation in G gives rise to an Fp -bilinear map δ : W × W → γ4 (G). We shall show that the maps δπi : W × W → γ4 (G)/Zi , for i = 1, 2, are non-degenerate. We compute ¯

¯

[ua v b [y, x]c [z, x]d , ua¯ v b [y, x]c¯ [z, x]d ] = ¯

¯

¯

= [y, x, u]c¯a−a¯c [z, x, v]db−bd [[z, x], [y, x]]d¯c−cd ¯

¯

¯

= [y, x, x, z](c¯a−a¯c)+(db−bd) [[z, x], [y, x]](d¯c−cd)−2(c¯a−a¯c) .

118


Thus the map δπ1 has matrix 

−1

    1



−1    

1 with respect to the given bases of W and γ4 (g)/Z1 ; it follows that the map δπ1 is non-degenerate. On the other hand, the map δπ2 has matrix 



s+2

    −s − 2

s −1 −s

   

1

with respect to the given bases of W and γ4 (G)/Z2 . This matrix has determinant s2 (s + 2)2 6≡ 0 (mod p), because s 6≡ 0, −2 (mod p). Hence δπ2 is non-degenerate. The non-degeneracy of δπ1 , δπ2 means that bg, hγ2 (Gi ), u, vic ⊇ Ni for all g ∈ hγ2 (Gi ), u, vi \ γ3 (Gi ), for i = 1, 2. In particular, it follows that bg, Gi c ⊇ Ni for all g ∈ hγ2 (Gi ), u, vi \ γ3 (Gi ), for i = 1, 2. We have proved that bg, Gi c ⊇ Ni , for all g ∈ Gi \ Ni , for i = 1, 2; hence all hypotheses of Corollary 4.3.2 have been verified, and its conclusion that G1 and G2 have identical character tables now follows. ¤

Chapter 7 Wreath products In this last chapter we shall study the character tables of wreath products. The theory of the characters of wreath products has been known for a long time. However, as far as we know, nobody has tried to isolate the exact ingredients on which the character table of a wreath product G o A depends. We shall show that such ingredients are the character table of G and the permutation group A: these determine the character table of G o A uniquely. What results is a powerful tool for increasing the derived length of a group, while keeping its character table under control. We shall employ it in Section 7.4 to construct pairs (G, H) of groups with identical character tables and derived lengths n and n + 1, for any given natural number n ≥ 2. It is a pleasure to thank Prof. I. M. Isaacs for the conversations on this subject which we had during his stay at the University of Warwick in June 1991.

7.1

Characters of wreath products

In this section and in the next one we shall collect some facts about irreducible characters, and respectively conjugacy classes of wreath products. Definition 7.1.1 Let G be a group and let A be a (not necessarily transitive) permutation group on a finite set Ω. Then the wreath product Γ = G o A is defined as the semidirect product [B]A, where the so-called base-group B = Q ω∈Ω Gω is the direct product of |Ω| = k copies of G and the action of A on 119

120

CHAPTER 7. WREATH PRODUCTS

B is given by (g1 , . . . , gk )a = (g1a−1 , . . . , gka−1 ) for all (g1 , . . . , gk ) ∈ B and for all a ∈ A. We shall use the notation (g1 , . . . , gk ), assuming that the set Ω is identified with the set {1, . . . , k}. Let us recall that the irreducible characters of the base group B = G1 × · · · × Gk , or more generally of any direct product G1 × · · · × Gk , where G1 , . . . , Gk are arbitrary groups, are exactly the characters θ = θ1 × · · · × θk with θi ∈ Irr(Gi ) for i = 1, . . . , k, where by definition θ(g1 , . . . , gk ) = θ1 (g1 ) · · · θk (gk ) for all (g1 , . . . , gk ) ∈ G1 × · · · × Gk . Furthermore, each θi is uniquely determined by θ, as the unique irreducible constituent of the restriction θGi . The action of A on B induces the following action of A on Irr(B): (θ1 × · · · × θk )a = θ1a−1 × · · · × θka−1 . In fact, we have −1

(θ1 × · · · × θk )a (g1 , . . . , gk ) = (θ1 × · · · × θk )((g1 , . . . , gk )a ) = (θ1 × · · · × θk )(g1a , . . . , gka ) = θ1 (g1a ) · · · θk (gka ) = θ1a−1 (g1 ) · · · θka−1 (gk ) = (θ1a−1 × · · · × θka−1 )(g1 , . . . , gk ). We shall employ the technique of tensor induction, an account of which was given in Section 2.5. A well-known result on characters of wreath products asserts that any irreducible character θ of the base group B which is invariant in Γ = [B]A is extendible to a character of Γ (a generalization of this fact is given in [14, Theorem 5.2]). In lemma 7.1.3 we shall compute explicitly an extension η ∈ Irr(Γ) of θ, but first let us do this in the special case in which A is cyclic and acts regularly on Ω. This special case is easier to prove and illustrates very well how tensor induction comes into play.

7.1. CHARACTERS OF WREATH PRODUCTS

121

Lemma 7.1.2 Let G be a group and let C be a cyclic group of order k. Let us form the regular wreath product Γ = G o C (that is to say, with respect to the regular permutation representation of C). Let θ = θ1 × · · · × θk be an irreducible character of the base group B and assume that θ is invariant in Γ. Then θ is extendible to Γ and an extension is given by ψ ⊗Γ , where ψ is the irreducible character of B defined by ψ = θ1 × 1G × · · · × 1G . Furthermore, if c is any generator of C, we have ψ ⊗Γ (c) = θ1 (1) = θ(1)1/k . Proof The base group B is a direct product of p copies of G. We may fix a generator c of C and assume that the action of C on B by conjugation is given by (g1 , . . . , gk )c = (gk , g1 , . . . , gk−1 ) for all (g1 , . . . , gk ) ∈ B. If θ is any irreducible character of B, then θ can be written in a unique way as θ1 × · · · × θk , where θ1 , . . . , θk are irreducible characters of G. The action of C on B induces the following action of C on Irr(B) (θ1 × · · · × θk )c = θk × θ1 × · · · × θk−1 . Let us assume now that θ = θ1 × · · · × θk is invariant in Γ; hence θc = θ. It follows that θ1 = θ2 = · · · = θk , and in particular that θ1 , . . . , θk have all the same degree, namely θ1 (1) = θ(1)1/k . Let us define ψ = θ1 × 1G × · · · × 1G and let us show that (ψ ⊗Γ )B = θ. For (g1 , . . . , gk ) ∈ B we have ψ ⊗Γ (g1 , . . . , gk ) =

k−1 Y

ψ(ci (g1 , . . . , gk )c−i )

i=0

=

k−1 Y

ψ(gi+1 , . . . , gk , g1 , . . . , gi )

i=0

=

k−1 Y

θ1 (gi+1 )

i=0

=

k−1 Y

θi+1 (gi+1 )

i=0

= θ(g1 , . . . , gk ),

122


where we have employed the formula for ψ ⊗Γ given by Lemma 2.5.2. Hence ψ ⊗Γ extends θ. To prove the last assertion, let ci be a generator of C. Since C = hci i acts transitively on the transversal T = C of B in Γ via ·, we have ψ ⊗Γ (ci ) = ψ(cik ) = ψ(1) = θ1 (1). The proof is complete.

¤

The general form of Lemma 7.1.2 is the following. Lemma 7.1.3 Let G be a group and let A be a permutation group on Ω, with |Ω| = k. Let us form the wreath product Γ = G o A. Let θ = θ1 × · · · × θk be an irreducible character of the base group B, and let us assume that θ is invariant in Γ. Then θ has an extension η ∈ Irr(Γ), and the value of η on the generic element g = (g1 , . . . , gk )a of Γ is given by the formula (7.1)

η(g) =

l Y i=1

θωi (gωi gωia · · · gωan(i)−1 ), i

where ω1 , . . . , ωl are representatives for the orbits of hai on Ω and n(i) is the length of the hai-orbit containing ωi . Proof Let us decompose Ω into A-orbits ˙ v. Ω = Ω1 ∪˙ · · · ∪Ω We may identify Ω with the set {1, . . . , k} in such a way that Ω1 = {k1 = 1, . . . , k2 − 1} Ω2 = {k2 , . . . , k3 − 1} .. . Ωv = {kv , . . . , k}. Then k1 , . . . , kv is a set of representatives for the orbits of A on Ω. Let Si denote the stabilizer of ki in A, for i = 1, . . . , v. By assumption θ is invariant in Γ. This is equivalent to (θ1 × · · · × θk )a = θ1 × · · · × θk for all a ∈ A,


123

that is to say, θ1a−1 × · · · × θka−1 = θ1 × · · · × θk for all a ∈ A. Since the components θj are uniquely determined by θ, we have θj a = θj for all j = 1, . . . , k and for all a ∈ A. Now let us define the following characters of B, for i = 1, . . . , v: θî = 1G1 × · · · × 1Gki −1 × θki × · · · × θki+1 −1 × 1Gki+1 × · · · × 1Gk . Each θî is irreducible and is clearly invariant in Γ. We shall find an extension ηi ∈ Irr(Γ) of θî for each i = 1, . . . , v. Since θ = θˆ1 · · · θˆv , the character η = η1 · · · ηv of Γ will be an extension of θ. Let us fix an index i. The stabilizer Si of ki in A centralizes Gki ; consequently, the subgroup BSi of Γ decomposes into a direct product   Y

 BSi = Gki ×  









 Gj   Si  .

1≤j≤k j6=ki

Thus there exists a unique extension ψi ∈ Irr(BSi ) of the irreducible character θki of Gki , such that 



 Y   Gj    Si ≤ ker ψi . 1≤j≤k j6=ki

The value of ψi on a generic element g = (g1 , . . . , gk )a of BSi is given by ψi (g) = θki (gki ). Now we claim that the tensor induced character ηi = ψi⊗Γ is an extension of θî . In order to prove the claim, let Ri be a set of representatives for the right cosets of Si in A; hence Ri also represents the right cosets of BSi in Γ. The action of Γ by right translation on the set of right cosets of BSi in Γ

124


induces an action of Γ on Ri , namely r · g for r ∈ Ri and g ∈ Γ is the unique element of Ri such that (BSi r)g = BSi (r · g). Let us fix an element g = (g1 , . . . , gk )a of Γ. Let Ri0 be a set of representatives of the orbits of hgi on R via ·, and for r ∈ Ri0 let n(r) denote the length of the hgi-orbit rhgi . According to Lemma 2.5.2, we have ψi⊗Γ (g) =

Y

ψi (rg n(r) r−1 ).

r∈Ri0

We compute: g n(r) = ((g1 , . . . , gk )a)n(r) = (g1 , . . . , gk )(g1 , . . . , gk )a

−1

· · · (g1 , . . . , gk )a

−(n(r)−1)

an(r)

= (g1 , . . . , gk )(g1a , . . . , gka ) · · · (g1an(r)−1 , . . . , gkan(r)−1 )an(r) = (g1 g1a · · · g1an(r)−1 , . . . , gk gka · · · gkan(r)−1 )an(r) . It follows that rg n(r) r−1 = (g1r g1ra · · · g1ran(r)−1 , . . . , gkr gkra · · · gkran(r)−1 )(ran(r) r−1 ). According to the definition of n(r), we have r · g n(r) = r, which means BSi rg n(r) = BSi r, or in other words rg n(r) r−1 ∈ BSi . Hence ran(r) r−1 ∈ A ∩ BSi = Si . Therefore we have ψi (rg n(r) r−1 ) = θki (gkir gkira · · · gkran(r)−1 ) i

for r ∈ Ri0 ; it follows that ψi⊗Γ (g) =

Y r∈Ri0

θki (gkir gkira · · · gkran(r)−1 ). i

We said earlier that θj a = θj for all j = 1, . . . , k and for all a ∈ A; hence we may also write (7.2)

ψi⊗Γ (g) =

Y r∈Ri0

θkir (gkir gkira · · · gkran(r)−1 ). i


125

We shall see that this result can be formulated differently. Let g = (g1 , . . . , gk )a ∈ Γ. Then g acts on Ri via · as a does; in fact, we have −1

(BSi r)g = (BSi (g1 , . . . , gk )r )ra = (BSi r)a, and thus r · g = r · a for all r ∈ Ri . Since Ri0 is a set of representatives of the orbits of hgi on Ri via ·, it is also a set of representatives of the orbits of hai on Ri via ·. Now the action of A on Ri via · is similar to the given permutation representation of A on Ωi , because Ri is a right transversal for the stabilizer Si of ki in A. More precisely, the map Ri → Ωi r 7→ kir is a bijection and satisfies kir·a = (kir )a . Therefore, the elements kir for r ∈ Ri0 are distinct, they form a set of representatives for the orbits of hai on Ωi , and n(r) is the length of the hai-orbit containing kir . Thus formula 7.2 can be rewritten as follows: (7.3)

ψi⊗Γ (g)

=

li Y j=1

θωij (gωij gωija · · · gωan(i,j)−1 ), ij

where ωi1 , . . . , ωili are representatives for the orbits of hai on Ωi , and n(i, j) is the length of the hai-orbit which contains ωij . In the particular case in which g = (g1 , . . . , gk )a ∈ B, we have a = 1, and hence all orbits of hai on Ωi have length one. Thus we get ψi⊗Γ (g) =

ki+1 −1

Y

θj (gj ) = θî (g).

j=ki

Hence ηi = ψi⊗Γ is an extension of θî , as we claimed. Let us define η = η1 · · · ηv . Then η extends θ, in fact ηB = (η1 )B · · · (ηv )B = θˆ1 · · · θˆv = θ.

126


Moreover, the value of η on a generic element g = (g1 , . . . , gk )a of Γ is given by the formula η(g) = η1 (g) · · · ηv (g) =

li v Y Y i=1 j=1

θωij (gωij gωija · · · gωan(i,j)−1 ), ij

where ωi1 , . . . , ωili are representatives for the orbits of hai on Ωi , and n(i, j) is the length of the hai-orbit containing ωij . The above formula can clearly be written as η(g) =

l Y i=1

θωi (gωi gωia · · · gωan(i)−1 ), i

where ωi , . . . , ωi are representatives for the orbits of hai on Ω and n(i) is the length of the hai-orbit containing ωi . The proof is complete. ¤ We observe that the formula for the extension η of θ given in Lemma 7.1.3 does not contain any trace of the orbits of A on Ω, which instead are fundamental in the proof of the lemma. In order to compute η(g) for g = (g1 , . . . , gk )a ∈ Γ, we only need to know the action of hai on B. Since the subgroup Bhai of Γ is naturally isomorphic to Gohai, where hai is regarded as a permutation group on Ω as a subgroup of A, it follows that in order to compute η(g) we may as well apply the lemma with A = hai. This leads to a sort of ‘canonicity’ of the extension η of θ, as stated in the next corollary. Corollary 7.1.4 Let A be a permutation group on a set Ω and let A1 be a subgroup of A; hence A1 is also a permutation group on Ω. Let G be a group and let us form the wreath product Γ = G o A. Let Γ1 be the subgroup of Γ generated by the base group B of Γ and A1 ; hence Γ1 is naturally isomorphic to G o A1 . Let θ ∈ Irr(B) be invariant in Γ, and let η, η1 be the extensions of θ to Γ and Γ1 respectively, constructed as in Lemma 7.1.3. Then ηΓ1 = η1 . Proof The conclusion follows easily from the discussion which precedes the corollary. ¤

7.2. CONJUGACY CLASSES OF WREATH PRODUCTS

7.2

127

Conjugacy classes of wreath products

Now that we have an explicit way of extending invariant characters of the base group of a wreath product Γ = G o A, let us turn our attention to the conjugacy classes of Γ. We shall partition Γ into subsets which are not conjugacy classes, though each of them is contained in some conjugacy class of Γ. The result of Lemma 7.1.2 would suggest to associate a subset Ka,(L1 ,... ,Ll ) of Γ to each a ∈ A and each l-ple (L1 , . . . , Ll ) of conjugacy classes of G, where l is the number of orbits of hai on Ω, according to the following definition: Ka,(L1 ,... ,Ll ) = {(g1 , . . . , gk )a ∈ Γ | gω1 gω1 a · · · gω1 an(1)−1 ∈ L1 , .. . gωl gωl a · · · gω an(l)−1 ∈ Ll }, l

where ω1 , . . . , ωl are representatives for the orbits of hai on Ω and n(i) is the length of the hai-orbit containing ωi (it can be easily shown that this definition is independent of the choice of representatives ω1 , . . . , ωl ). In fact, if θ is an irreducible character of the base group B which is invariant in Bhai, then the extension η of θ given by the formula of Lemma 7.1.2 is clearly constant on Ka,(L1 ,... ,Ll ) . We shall define the subsets Ka,(L1 ,... ,Ll ) of Γ using a slightly different notation which will allow us to describe more easily how A permutes them by conjugation. Let M denote the set of maps m : Ω → cl(G), where cl(G) is the set of conjugacy classes of G. The action of A on Ω induces an action of A on M, where ma for m ∈ M and a ∈ A is the map such that −1

(ma )(ω) = m(ω a ) for all ω ∈ Ω. Let us define Ma for a ∈ A as the subset of the elements of M fixed by a; in other words Ma is the set of maps m : Ω → cl(G) which are constant on the orbits of hai on Ω. We observe that (Ma )b for a, b ∈ A is the set of the elements of M fixed by ab ; hence (Ma )b = Mab .

128


Definition 7.2.1 Let a ∈ A and m ∈ Ma . Let us choose representatives ω1 , . . . , ωl for the orbits of hai on Ω and let n(i) be the length of the hai-orbit containing ωi . Let us define Ka,m = {(g1 , . . . , gk )a ∈ Γ | gω1 gω1 a · · · gω1 an(1)−1 ∈ m(ω1 ), .. . gωl gωl a · · · gω an(l)−1 ∈ m(ωl )}. l

This definition does not depend on the choice of the representatives ω1 , . . . , ωl of the orbits of hai on Ω. In fact, let us replace for example m ωi with a different representative ω ¯ i of its hai-orbit, hence ω ¯ i = ωia for some integer m with 1 ≤ m ≤ n(i) − 1. The element gω¯ i gω¯ ia · · · gω¯ an(i)−1 = gωi am gωi am+1 · · · gω am+n(i)−1 i

i

−1

= (gωi gωi a · · · gωi am−1 ) (gωi gωi a · · · gωi am−1 ) (gωi am · · · gω an(i)−1 )(gω an(i) · · · gω am+n(i)−1 ) i

i

i

−1

= (gωi gωi a · · · gωi am−1 ) (gωi gωi a · · · gω an(i)−1 )(gωi gωi a · · · gωi am−1 ) i

is conjugate to gωi gωi a · · · gω an(i)−1 in G, hence it belongs to the conjugacy i class m(¯ ωi ) = m(ωi ) exactly when gωi gωi a · · · gω an(i)−1 does. Therefore, the i definition of the set Ka,m is independent of the choice of the representatives ω1 , . . . , ω l . It is clear that the sets Ka,m for a ∈ A and m ∈ Ma form a partition of Γ. In its action on Γ by conjugation A permutes the sets Ka,m , as the next lemma states. Lemma 7.2.2 For a, b ∈ A and m ∈ Ma , we have (Ka,m )b = Kab ,mb . Proof First of all, the elements of Ka,m have the form (g1 , . . . , gk )a for some g1 , . . . , gk ∈ G. Similarly, the elements of Kab ,mb have the form (g1 , . . . , gk )ab for some g1 , . . . , gk ∈ G. Let ω1 , . . . , ωl be representatives for the orbits of hai on Ω, then ω1b , . . . , ωlb are representatives for the orbits of hab i on Ω

7.2. CONJUGACY CLASSES OF WREATH PRODUCTS

129

and the hab i-orbit containing ωib has the same length n(i) of the hai-orbit containing ωi . We have ((g1 , . . . , gk )a)b = (g1b−1 , . . . , gkb−1 )ab = (h1 , . . . , hk )ab , where we have put hj = gj b−1 for all j = 1, . . . , k. The condition (h1 , . . . , hk )ab ∈ Kab ,mb

(7.4) is equivalent to

hωi b hωi b(ab ) · · · hω b(ab )n(i)−1 ∈ mb (ωib ) for all i = 1, . . . , l. i

Since hωi b hωi b(ab ) · · · hω b(ab )n(i)−1 = hωi b hωi ab · · · hω an(i)−1 b i

i

= gωi gωi a · · · gω an(i)−1 , i

and mb (ωib ) = m(ωi ) by definition, condition 7.4 is equivalent to gωi gωi a · · · gω an(i)−1 ∈ m(ωi ) for all i = 1, . . . , l, i

and hence to (g1 , . . . , gk )a ∈ Ka,m , that is to say, ((g1 , . . . , gk )a)b ∈ (Ka,m )b . The proof is complete.

¤

In the next lemma we shall compute the cardinality of the sets Ka,m . Lemma 7.2.3 We have |Ka,m | = |G|

k−l

l Y

|m(ωi )|,

i=1

where ω1 , . . . , ωl are representatives for the orbits of A on Ω.

130


Proof We observe that the equation ˆ h1 h2 · · · hn = h ˆ is a fixed element of a group H, in the unknowns h1 , . . . , hn ∈ H, where h n−1 has exactly |H| solutions (h1 , . . . , hn ). In fact, given arbitrary values in H −1 ˆ to h1 , . . . , hn−1 , there is exactly one value for hn , namely hn = h−1 n−1 · · · h1 h, ˆ Now, from the definition of Ka,m we easily get such that h1 h2 · · · hn = h. |Ka,m | =

l Y

(|G|n(i)−1 |m(ωi )|),

i=1

where n(i) is the length of the hai-orbit containing ωi . Since the conclusion follows.

7.3

Pl

i=1

n(i) = k, ¤

Character tables of wreath products

Theorem 7.3.1 Let G1 and G2 be groups with identical character tables and let A be a permutation group on Ω. Then the wreath products Γ1 = G1 o A and Γ2 = G2 o A have identical character tables. Proof Let G1 and G2 have identical character tables via the bijections α ˜ : G1 → G2 , β˜ : Irr(G1 ) → Irr(G2 ). We shall prove the theorem by defining bijections α : Γ1 → Γ2 , β : Irr(Γ1 ) → Irr(Γ2 ), and then checking that χβ (g α ) = χ(g) for all χ ∈ Irr(Γ1 ) and for all g ∈ Γ1 . Definition of α.

7.3. CHARACTER TABLES OF WREATH PRODUCTS

131

Let us define subset of Γ1 and Γ2 according to Definition 7.2.1. Since now we have two wreath products Γ1 = G1 o A and Γ2 = G2 o A, we shall keep the notation of Definition 7.2.1 for what concerns the group Γ1 , and add bars for ¯ a for a ∈ A will the corresponding objects of Γ2 . In particular, Ma and M denote the set of maps m : Ω → cl(G1 ), and respectively m ¯ : Ω → cl(G2 ), which are constant on each orbit of hai on Ω. For each m ∈ M let us define a map mα˜ : Ω → cl(G2 ), that is to say, an ¯ via the formula element mα˜ of M, mα˜ (ω) = m(ω)α˜ for all ω ∈ Ω. ¯ defined above commutes with the We observe that the map α ˜ : M → M ¯ namely actions of A on M and M, (mα˜ )a = (ma )α˜ for all a ∈ A. In fact, for ω ∈ Ω we have −1

−1

(mα˜ )a (ω) = mα˜ (ω a ) = m(ω a )α˜ = ma (ω)α˜ = (ma )α˜ (ω). ¯ a exactly when m ∈ Ma ; As a consequence, for a ∈ A we have that mα˜ ∈ M in other words, for each a ∈ A we get a bijection ¯a Ma → M m 7→ mα˜ . Now the sets Ka,m for a ∈ A and m ∈ Ma form a partition of Γ1 . ¯ a,mα˜ for a ∈ A and m ∈ Ma form a partition of Γ2 . Similarly, the sets K Furthermore, according to Lemma 7.2.3, we have |Ka,m | = |G1 |k−l

l Y

|m(ωi )|,

i=1

and ¯ a,mα˜ | = |G2 |k−l |K

l Y

|mα˜ (ωi )|,

i=1

˜ where ω1 , . . . , ωl are representatives for the orbits of hai on Ω. Since (˜ α, β) is a character table isomorphism, we have |G1 | = |G2 | and |m(ωi )| = |m(ωi )α˜ | = |mα˜ (ωi )|.

132


It follows that ¯ a,mα˜ | for all a ∈ A and for all m ∈ Ma . |Ka,m | = |K ¯ a,mα˜ for Thus we can choose a bijection α : Γ1 → Γ2 which sends Ka,m onto K all a ∈ A and for all m ∈ Ma . Definition of β. The bijection β˜ : Irr(G1 ) → Irr(G2 ) induces a bijection β¯ : Irr(B1 ) → ¯ Irr(B2 ), where θβ for θ = θ1 × · · · × θk ∈ Irr(B1 ) is defined as ¯

˜

˜

θβ = θ1β × · · · × θkβ ∈ Irr(B2 ). The bijection β¯ commutes with the action of A by ‘conjugation’ on Irr(B1 ) and Irr(B2 ), namely ¯

¯

(θa )β = (θβ )a for all θ ∈ Irr(B1 ) and for all a ∈ A. In fact, if θ = θ1 × · · · × θk ∈ Irr(B1 ) we have ¯

˜

¯

˜

¯

(θa )β = (θ1a−1 × · · · × θka−1 )β = θ1βa−1 × · · · × θkβa−1 = (θβ )a . Let us choose representatives θ1 , . . . , θr for the orbits of A on Irr(B1 ). Then ¯ ¯ θ1β , . . . , θrβ are representatives for the orbits of A on Irr(B2 ). For i = 1, . . . , r let Ti denote the inertia group of θi in A, that is to say, the stabilizer of θi ¯ in the action of A on Irr(B1 ). Clearly, Ti is also the inertia group of θiβ in A. According to Clifford’s Theorem, Irr(Γ1 ) and Irr(Γ2 ) decompose as follows Irr(Γ1 ) = Irr(Γ1 , θ1 )∪˙ · · · ∪˙ Irr(Γ1 , θr ), ¯

¯ Irr(Γ2 ) = Irr(Γ2 , θ1β )∪˙ · · · ∪˙ Irr(Γ2 , θrβ ).

We shall define bijections ¯

β : Irr(Γ1 , θi ) → Irr(Γ2 , θiβ ), for i = 1, . . . , r, which will then be put together to give a bijection β : Irr(Γ1 ) → Irr(Γ2 ). Let us fix an index i. According to the Clifford correspondence (see [13, Theorem (6.11)]), induction of characters gives bijections from Irr(B1 Ti , θi )

7.3. CHARACTER TABLES OF WREATH PRODUCTS ¯

133

¯

onto Irr(Γ1 , θi ) and from Irr(B2 Ti , θiβ ) onto Irr(Γ2 , θiβ ). The construction of ¯ our bijection βi will thus pass through the sets Irr(B1 Ti , θi ) and Irr(B2 Ti , θiβ ). Now since θi is invariant in B1 Ti (which is canonically isomorphic to G1 o Ti ), Lemma 7.1.2 guarantees that θi is extendible to B1 Ti and provides us with a standard extension of θi , let us call it ηi ∈ Irr(B1 Ti ). Similarly, ¯ let us call ηî ∈ Irr(B2 Ti ) the standard extension of θiβ provided by Lemma 7.1.2. According to [13, Corollary (6.17)], the elements of Irr(B1 Ti , θi ) are exactly the characters ηi ϕ for ϕ ∈ Irr(B1 Ti /B1 ). Similarly, the elements of ¯ Irr(B2 Ti , θiβ ) are the characters ηî ψ for ψ ∈ Irr(B2 Ti /B2 ). We have a natural bijective correspondence between Irr(B1 Ti /B1 ) and Irr(B2 Ti /B2 ), corresponding to the obvious isomorphism from B1 Ti /B1 onto B2 Ti /B2 . To put it differently, the restriction map gives bijections Irr(Bj Ti /Bj ) → Irr(Ti ) ϕ 7→ ϕTj , for j = 1, 2; hence we can form a bijection Irr(B1 Ti /B1 ) → Irr(B2 Ti /B2 ) ϕ 7→ ϕ, ˆ where ϕˆ denotes the unique irreducible character of B2 Ti whose kernel contains B2 and such that ϕˆTi = ϕTi . ¯ Now we are ready to set up a bijection from Irr(Γ1 , θi ) onto Irr(Γ2 , θiβ ). In fact, we have seen that Irr(Γ1 , θi ) = {(ηi ϕ)Γ1 | ϕ ∈ Irr(B1 Ti /B1 )} and

¯

Irr(Γ2 , θiβ ) = {(ˆ ηi ϕ) ˆ Γ2 | ϕ ∈ Irr(B1 Ti /B1 )}.

Let us define the following maps, for i = 1, . . . , r: ¯

βi : Irr(Γ1 , θi ) → Irr(Γ2 , θiβ ) (ηi ϕ)Γ1 7→ (ˆ ηi ϕ) ˆ Γ2 . The maps βi are well defined and are bijections. The various maps βi can then be put together to give a single bijection β : Irr(Γ1 ) → Irr(Γ2 ).

134


Verification that χβ (g α ) = χ(g). Let us fix a character χ ∈ Irr(Γ1 ), say χ ∈ Irr(Γ1 , θi ). Then χ = (ηi ϕ)Γ1 for some ϕ ∈ Irr(B1 Ti /B1 ), where ηi is the standard extension of θi to B1 Ti given by Lemma 7.1.2. According to our definition of β, we have χβ = (ˆ ηi ϕ) ˆ Γ2 , where ηî is the standard extension of θi to B2 Ti , and ϕˆ is the unique character in Irr(B2 Ti /B2 ) such that ϕˆTi = ϕTi . We shall first show that (ˆ ηi ϕ)(g ˆ α ) = (ηi ϕ)(g) for all g ∈ B1 Ti . In order to prove this fact, let us fix g = (g1 , . . . , gk )a ∈ B1 Ti . Then there is a unique m ∈ Ma such that g ∈ Ka,m , and hence we have gω1 gω1 a · · · gω1 an(1)−1 ∈ m(ω1 ), .. . gωl gωl a · · · gω an(l)−1 ∈ m(ωl ), l

where ω1 , . . . , ωl are representatives for the orbits of hai on Ω, and n(i) denotes the length of the hai-orbit which contains ωi . According to our ¯ a,mα˜ , and hence g α = (h1 , . . . , hk )a for some definition of α, we have g α ∈ K h1 , . . . , hk ∈ G2 , such that hω1 hω1 a · · · hω1 an(1)−1 ∈ mα˜ (ω1 ), .. . hωl hωl a · · · hω an(l)−1 ∈ mα˜ (ωl ). l

Let us put θi = θi1 × · · · × θik . According to Lemma 7.1.2, we have ηi (g) =

l Y j=1

θiωj (gωj gωj a · · · gω

j

an(j)−1

),

and similarly, ηî (g α ) =

l Y

˜

(θiωj )β (hωj hωj a · · · hω

j=1

j

an(j)−1

).

7.3. CHARACTER TABLES OF WREATH PRODUCTS

135

Now for each j = 1, . . . , l we have gωj gωj a · · · gω

j

an(j)−1

∈ m(ωj ),

and hωj hωj a · · · hω

j

an(j)−1

∈ mα˜ (ωj ) = m(ωj )α˜ .

˜ is a character table isomorphism, it follows that From the fact that (˜ α, β) (θiωj )β (hωj hωj a · · · hω

j

an(j)−1

) = θiωj (gωj gωj a · · · gω

j

an(j)−1

)

for all j = 1, . . . , l. Hence we have ηî (g α ) = ηi (g). Since we also have ϕ(g ˆ α ) = ϕ(a) ˆ = ϕ(a) = ϕ(g), it follows that (ˆ ηi ϕ)(g ˆ α ) = (ηi ϕ)(g) for all g ∈ B1 Ti , as claimed. Finally, we shall show that χβ (g α ) = χ(g) for all g ∈ Γ1 . Let (ηi ϕ)◦ (respectively (ˆ ηi ϕ) ˆ ◦ ) denote the function on Γ1 (respectively Γ2 ) which extends ηi ϕ (respectively ηî ϕ) ˆ and vanishes on Γ1 \ B1 Ti (respectively Γ2 \ B2 Ti ). We clearly have (ˆ ηi ϕ) ˆ ◦ (g α ) = (ηi ϕ)◦ (g) for all g ∈ Γ1 . Let us fix g ∈ Γ1 ; we compute χ(g) = (ηi ϕ)Γ1 (g) = =

X 1 (ηi ϕ)◦ (xgx−1 ) |B1 Ti | x∈Γ1

1 X (ηi ϕ)◦ (bgb−1 ). |Ti | b∈A

Similarly, we have χβ (g α ) = (ˆ ηi ϕ) ˆ Γ2 (g α ) = =

X 1 (ˆ ηi ϕ) ˆ ◦ (xg α x−1 ) |B2 Ti | x∈Γ2

1 X (ˆ ηi ϕ) ˆ ◦ (bg α b−1 ). |Ti | b∈A

136


In order to conclude that χβ (g α ) = χ(g) for all g ∈ Γ1 , it will be enough to show that (ˆ ηi ϕ) ˆ ◦ (bg α b−1 ) = (ηi ϕ)◦ (bgb−1 ). Since we have already proved that (ˆ ηi ϕ) ˆ ◦ ((bgb−1 )α ) = (ηi ϕ)◦ (bgb−1 ), it remains to show that (ˆ ηi ϕ) ˆ ◦ (bg α b−1 ) = (ˆ ηi ϕ) ˆ ◦ ((bgb−1 )α ) ¯ c,m¯ , the equality above, and hence the Because (ˆ ηi ϕ) ˆ ◦ is constant on each K conclusion of the proof, will follow from the fact that bg α b−1 and (bgb−1 )α ¯ c,m¯ . To prove this fact, we observe that on one hand, belong to the same K α ¯ from g ∈ Ka,mα˜ we get −1

bg α b−1 = (g α )b

¯ b−1 α˜ b−1 , ∈K a ,(m )

according to Lemma 7.2.2. On the other hand, since g ∈ Ka,m , we have bgb−1 ∈ Kab−1 ,mb−1 , again according to Lemma 7.2.2; consequently, we obtain ¯ b−1 b−1 α˜ = K ¯ b−1 α˜ b−1 , (bgb−1 )α ∈ K a ,(m ) a ,(m ) −1

−1

because (mb )α˜ = (mα˜ )b . This concludes the proof.

7.4

¤

An application to character tables and derived length

In this last section we shall construct, for any given natural number n with n ≥ 2, a pair of groups G1 and G2 with identical character tables and derived lengths n and n + 1 respectively. Let us begin with a standard result.

7.4. AN APPLICATION

137

Lemma 7.4.1 Let G be a soluble group and C be a non-trivial cyclic group. Let us form the regular wreath product Γ = G o C. Then we have dl(Γ) = dl(G) + 1. Proof The base group B is the direct product B = G1 × · · · × Gr of r = |C| isomorphic copies of G. We may fix a generator c of C and assume that c acts on B as follows: (g1 , . . . , gr )c = (gr , g1 , . . . , gr−1 ) for all (g1 , . . . , gr ) ∈ B. We claim that Γ0 = {(g1 , . . . , gr ) ∈ B | g1 · · · gr ∈ G0 }. We observe that B 0 = G01 × · · · × G0r is a normal subgroup of Γ contained in Γ0 ; hence Γ0 /B 0 = (Γ/B 0 )0 . Since Γ/B 0 is canonically isomorphic to the regular wreath product (G/G0 ) o C, it will be enough to prove the claim with the additional assumption that G is abelian. Let us assume that G is abelian and put M = {(g1 , . . . , gr ) ∈ B | g1 · · · gr = 1}. Clearly, M is a subgroup of the abelian group B; furthermore, M is normal in Γ, because it is C-invariant. Let (h1 , . . . , hr ) ∈ B. We have −1 c [(h1 , . . . , hr ), c−1 ] = (h−1 1 , . . . , hr )(h1 , . . . , hr )

−1

−1 = (h−1 1 , . . . , hr )(h2 , . . . , hr , h1 ) −1 −1 = (h−1 1 h2 , . . . , hr−1 hr , hr h1 ) ∈ M.

It follows that Γ0 = (BC)0 = [B, C] ≤ M. Now let (g1 , . . . , gr ) ∈ M . Let us put h1 = 1 and hi+1 = hi gi for i = 1, . . . , r − 1. Then we have −1 −1 [(h1 , . . . , hr ), c−1 ] = (h−1 1 h2 , . . . , hr−1 hr , hr h1 ) −1 · · · g1−1 ) = (g1 , . . . , gr−1 , gr−1

= (g1 , . . . , gr ).

138


Thus we have M ≤ Γ0 . We conclude that Γ0 = M , and our claim is proved. In order to prove the lemma now it suffices to show that dl(Γ0 ) = dl(G). Since Γ0 ≤ B we have dl(Γ0 ) ≤ dl(B) = dl(G). On the other hand, the group homomorphism Γ0 → G (g1 , . . . , gr ) 7→ g1 is surjective, because the element (g, g −1 , 1, . . . , 1) of B is mapped to the generic element g of G. Hence dl(Γ0 ) ≥ dl(G). We conclude that dl(Γ0 ) = dl(G), and the lemma is proved. ¤ Theorem 7.4.2 Let n be a natural number with n ≥ 2. Then there exist groups G1 and G2 with identical character tables, such that G1 has derived length n, while G2 has derived length n + 1. Proof We shall prove the theorem by induction on n. For the case n = 2, the existence of groups G1 and G2 with identical character tables and derived lengths 2 and 3 respectively was proved in Chapters 5 and 6, where two different constructions were used. Now let us fix n > 2 and assume that we have been able to construct groups G1 and G2 with identical character tables and derived lengths n − 1 and n respectively. Let C be a non-trivial cyclic group. Let us form the regular wreath product Γi = Gi o C, for i = 1, 2. Then Γ1 and Γ2 have identical character tables, according to Theorem 7.3.1. On the other hand, the derived lengths of Γ1 and Γ2 are n and n + 1 respectively, according to Lemma 7.4.1. This concludes the proof. ¤ We observe that there are p-groups G1 and G2 which satisfy the conclusions of Theorem 7.4.2 (at least when the prime p is greater than or equal to 5); in fact, we may take the p-groups G1 and G2 constructed in Chapter 6 as the basis of the inductive proof of Theorem 7.4.2, and then take a cyclic group of order p as the group C of the induction step. We conclude this thesis with an open question.

7.4. AN APPLICATION

139

Question 7.4.3 Is there any pair (G, H) of groups which have identical character tables, and derived lengths two and four respectively?

Bibliography [1] B. Beisiegel, Semi-extraspeziellen p-gruppen, Math. Z. 156 (1977), 247– 254. [2] A. R. Camina, Some conditions which almost characterize Frobenius groups, Israel J. Math. 31 (1978), 153–160. [3] D. Chillag, I. D. Macdonald and C. M. Scoppola, Generalized Frobenius groups II, Israel J. Math. 62 (1988), 269–282. [4] J. H. Conway, T. R. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985. [5] C. W. Curtis and I. Reiner, Methods of Representation Theory I, Wiley Interscience, New York, 1981. [6] E. C. Dade, Answer to a Question of R. Brauer, J. of Algebra, 1 (1964), 1–4. [7] K. Dock and P. Igodt, Character tables and commutativity of normal subgroups, SIGSAM Bull. of ACM, 25 (1991), 28–31. [8] M. Hall, The theory of groups, Macmillan Company, New York, 1959. [9] G. Higman, Suzuki 2-groups, Illinois J. Math. 7 (1963), 79–96. [10] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967. [11] B. Huppert, N. Blackburn, Finite Groups II, Springer-Verlag, Berlin, Heidelberg, New York, 1982. 140

BIBLIOGRAPHY

141

[12] B. Huppert, R. Gow, R. Knörr, O. Manz, R. Staszewski, W. Willems, Lectures on “Clifford theory and applications”, Trento, September 14– 18, 1987, Centro Internazionale per la ricerca matematica, Trento. [13] I. M. Isaacs, Character theory of finite groups, Academic Press, New York, San Francisco, London, 1976. [14] I. M. Isaacs, Character Correspondences in Solvable Groups, Advances in Mathematics 43 (1982), 284–306. [15] N. Jacobson, Basic Algebra I, W. H. Freeman and company, San Francisco, 1973. [16] A. Mann and C. M. Scoppola, On p-groups of Frobenius type, Arch. Math. 56 (1991), 320–332. [17] S. Mattarei, Character tables and metabelian groups, J. London Math. Soc., to appear. [18] H. Neumann, Varieties of Groups, Springer-Verlag, Berlin, Heidelberg, New York, 1967. [19] T. Saaty (Editor), Lectures on Modern Mathematics, Volume 1, John Wiley & Sons, New York, 1963. [20] A. I. Saksonov, An answer to a question of R. Brauer (Russian), Vesci Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk (1967), no. 1, pp. 129–130 (Math. Reviews 35 (1968) ] 5527). [21] C. M. Scoppola, Groups of prime power order as Frobenius-Wielandt complements, Trans. Amer. Math. Soc. (2) 325 (1991), 855–874.

Retrieving information about a group from its character table

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