Permutation group theory and permutation polynomials

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Permutation group theory and permutation polynomials Article · January 1999

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133

Permutation Group Theory and Permutation Polynomials Stephen D Cohen * Department

of Mathematics, University of Glasgow Glasgow G12 8QW, Scotland [email protected]

Abstract We survey the connection between the theory of finite permutation groups (especially primitive groups) and an important type of permutation polynomial over a finite field K, namely, that comprising polynomials which permute infinitely many finite extensions of K. The groups arise as monodromy groups and the theory has led the way to the recent discovery of highly non-trivial families of permutation polynomials.

1

Introduction

Let K = GF(q) be the finite field of order q = p" and prime characteristic p. Throughout we use f to denote a monic polynomial in K[X] of degree n = plm (where p f m), l ~ 0, not of the form Then f is a permutation polynomial (PP) over K if, as a function, it permutes the elements of K, i.e., f(K) = K. Of course, any permutation of K can be represented by a PP. Nevertheless, there are not many families of easily recognisable PPs. Important known examples include certain families of exceptional polynomials. By definition, f is an exceptional polynomial (EP) over K if it is a PP over infinitely many finite extensions of K. (In particular, an EP over K is a PP over K). This is a strong condition; EPs, however, constitute some of the most useful classical examples of PPs (see Section 4). The purpose of this article is to survey is to survey a connection between EPs and the theory of primitive permutation groups. The source of this connection is the following. Let t be an indeterminate and L the splitting field of the (irreducible) polynomial f(X) - t over K(t). By the assumption f =j:. g,

g.

f(X)

- t = (X - xd ... (X - xn),

"The author is grateful to the Royal Society for a Conference Grant to attend the ICAC97 meeting in Hong Kong at which this paper was presented and to the organisers of ICAC97 for their hospitali ty.

134 where the roots by

Xl,

...

, Xn

(in L) are distinct.

A := Gal(f(X)

- t, K(t))

Let A be the Galois group defined = Gal(LI K(t)),

regarded as a (transitive) permutation group on the roots Xl, ... , Xn (and so having degree n). Then A is called the arithmetic monodromy group of f. Similarly, if k is the algebraic closure of the constant field K in L, we define the Galois group G, a normal subgroup of A, by G := Gal(f(X)

- t, k(t))

= Gal(LI k(t))

and call it the geometric monodromy group of f. Then the requirement that f be an EP can be cast as an equivalent condition on A and G (see Section 3). In particular, it is essential that K and tc (and so A and G) are different. Recall that AIG ~ Gal(kIK), a cyclic group. As far as existence questions are concerned, it suffices to look for EPs f that are (functionally) indecomposable, i.e., any decomposition f = g(h), where g and h are polynomials over K, has g or h linear. In group-theoretical terms, this is equivalent to the permutation group A being primitive. Thus, the theory of primitive permutation groups can be used. A key reference on the topic is the important paper by Fried, Guralnick and Saxl [11] which summarises material from the decade (around 1970) of early activity on EPs and proceeds to make serious use of group theory (including the classification of finite simple groups) plus other tools to restrict drastically the possible monodromy groups of an EP, see Section 6. This has led to the discovery of highly non-trivial examples of families of EPs by Cohen and Matthews, [6]' [7], Lenstra and Zieve [16] and Guralnick and Muller [12]. These are described in Sections 7 and 8. It is possible to relax the conditions we have assumed in various ways and still exploit group theory. We shall not discuss such extensions here, but these might include allowing f to be a rational function, or K to be an infinite field (of characteristic zero or a prime), assuming Gal(I{ I K) to be cyclic or otherwise, or considering some alternative concept to exceptionality. Throughout we shall retain the notation and assumptions of this introduction, except where mentioned otherwise.

2

Monodromy groups as permutation groups

Let A and G be the arithmetic and geometric monodromy groups of f as described in Section 1. Note that, if K is an algebraic closure of K, then = K n Land G = Gal(f(X) - t,K(t)). We can write Al for a I-point stabilizer of A and, more specifically, regard Al as the subgroup of A fixing the root Xl. Similarly, GI is used for a 1point stabilizer of G. One of the equivalent definitions for A to be a primitive permutation group is that Al is a maximal subgroup of A, [23]' §8.

tc

I

135

As remarked in Section 1, there is a connection between indecomposable polynomials over J( and primitivity. It is recorded for example in [10] Proposition is a primitive

2.1 The polynomial group.

f is indecomposable

over J( if and only if A

We next summarise some important consequences of the simple assumption that f is a polynomial. These are described in detail in [11] Because f is a polynomial then f(X) - t is "totally ramified at 00". This means that G has a solvable subgroup Goo (called the inertia group of a place of L over infinity) with properties such as those given in the next results, taken from [11]. The first is referred to as the "Goo-Iemma". Recall the notation n = ptm, where p f m and f 2': o. Lemma 2.2 The group Goo is a transitive subgroup of G with a (unique) normal p-Sylow subgroup Hoo. Further GooI Hoo is cyclic of order m*, where m divides m" but p does not divide m", In particular, if p does not divide n (so that n = m), then m* = m = nand Goo is generated by an n-cycle.

By a factorization of a group I' is meant its representation as a set-theoretic product I' = r1.r2. By the transitivity of Goo, we have the following. Corollary 2.3 The arithmetic monodromy group possesses the factorization A A1.Goo. Moreover, Al is maximal when f is indecomposable.

=

Considering the arithmetic monodromy group, we obtain a subgroup Aoo, the decomposition group of a place of Lover 00, containing Goo as a normal subgroup. The next result is not difficult. Lemma 2.4 The subgroup Hoo is actually a normal subgroup of A. Moreover,

3

Exceptional polynomials

As defined in Section 1, the polynomial f is an EP over J( if it is a pp over infinitely many finite extensions of J(. In fact, the "exceptional" nature of these polynomials was originally delineated (by Davenport and Lewis, [8]) in respect of another property which turns out to be equivalent (Proposition 3.1). In addition to [8]' basic facts about EPs are given in articles by MacCluer, [20], Hayes, [15], Cohen, [2] and Fried [10] (among others). Although cast in a somewhat different form, we summarise some results drawn from these papers in Propositions 3.13.4, below. Their proofs are intertwined. For instance, that of Proposition 3.1 depends on (parts of) the other results.

----------

136 The first of the results also conceals an appeal to (at least a weak form of) the Hasse- Weil theorem, [17), P 330, to apply which, q must be sufficiently large. In its statement involving Dickson polynomials (though more complicated than that in Theorem 7.1) has been obtained by M Zieve. In work in preparation, R Guralnick and M Zieve claim that all almost simple EPs derive from the MCM and LZ polynomials. It is clear that, for even k(~ 2), there exist analogues of the LZ polynomials (having F(A) = PSL2(q)) though, evidently, some modification of their definition is required.

8

Affine exceptional polynomials

Because of their irrelevance to the Carlitz and Wan conjectures, less progress has been made on the classification of affine EPs (satisfying Theorem 6.1(ii)). Nevertheless, examples beyond the classical ones (which have H cyclic) are of great potential interest. This is confirmed by the significant new examples reported below. The degree of an affine EP over K = GF(p'·) is of the form n = pi(e ~ 1). When n = p, it is not hard to see that the only EPs are classical (of type II). The next stage is to consider n = p2. In this case, group theory [13], [14) yields the following possibilities. Lemma 8.1 When n = p2, then an EP over K either is classical, or is such that p is odd and the one-point stabilizer G1 is dihedral.

With Lemma 8.1 as the spur, Guralnick and Muller [12) consider a more general situation. With some modification of our standard notation, set n = q2, where q = pi and let K = GF(qd) for d ~ 1. Then Guralnick and Muller exhibit a remarkable parametric family of polynomials over K. There are many choices of the parameter /.L in K which yield EPs. The result is as follows.

144

Theorem

8.2 Let q

K = GF(qd),

d

= pI,

where p is an odd prime

and

e

> 1,

and let

2: 1. For any p,(:j: 0) in K, set

where al" =

Then the following

x2q - 2p,Xq+l

+ 2p,Xq + p,2 X2 + 2p,2X + p,2.

hold.

• hI" is a polynomial

of degree q2 over K.

• hI" is an EP if and only if p, is not a (q - l)-th power in K. • A is a subgroup of the affine linear group AGL2(q).

• Cl is dihedral of order 2(q

+ 1).

• The orbits of Cl all have length q

+ 1.

• The lengths of the orbits of Al all exceed q

+ 1.

The construction of the polynomials hI" was by forcing them to have the right ramification data to fit the designed monodromy groups. To establish exceptionality, the authors did not factorize 4>, but did derive information on 1 the degrees of the irreducible factors of 4> over K and over K (p, ~) equivalent to the assertions of the theorem as regards orbit lengths. Note that, as p, varies, we obtain a family hI"' whose members are less obviously related than the twists of Theorems 7.1 and 7.2.

9

Inverse Galois theory

It is evident from Sections 7 and 8 that not only does group theory aid the search for and cast light on the nature of the relevant polynomials, but so also the manner in which the polynomials and their factorization, etc., represent the groups make the structure of the latter more transparent. Similarly, S S Abhyankar has observed connections between the kind of polynomials investigated with regard to exceptionality and group theory. In particular, he has a long-term project on discovering "nice equations for nice groups" and has written a number of papers on such a theme, expecially with respect to the classical linear groups. We quote an example from [I] in which the MCM polynomials jj, (k odd or even) feature. Take K = GF(2) and q = 2k. Then Xq2

+ t2 X" + X + t

= (fk(X)

+ t)(xq !k(X) + t.X" + 1)

has Galois group PSL2(q) = PSP2(Q) ~ P D3(q) over K(t). Here PSp and P D refer to the projective symplectic and orthogonal groups, respectively. See [1], Section 8, for further details of context.

145

References [1] S S Abhyankar, Factorizations over finite fields, Finite Fields Applications, London Math. Soc., Lect. Note Ser. 233 (1996), 1-21.

and

[2] S D Cohen, The distribution of polynomials over finite fields, Acta Arith. 17 (1970), 255-271. [3] S D Cohen, Exceptional polynomials and the reducibility of substitution polynomials, Enseign. Math. 36 (1990), 53-65. [4] S D Cohen, Permutation polynomials and primitive permutation groups, Arch. Math. (Easel) 57 (1991),417-423. [5] S D Cohen and M D Fried, Lenstra's proof of the Carlitz- Wan conjecture: an elementary version, Finite Fields Appl. 1 (1995), 372-375. [6] S D Cohen and R W Matthews, A class of exceptional polynomials, Trans. Amer. Math. Soc. 345 (1994), 897-909. [7] S D Cohen and R W Matthews, Exceptional polynomials over finite fields, Finite Fields Appl. 1 (1995), 261-277. [8] H Davenport and D J Lewis, Notes on congruences, I, Quart. J. Math. Oxford Ser. (2) 14 (1963), 51-60. [9] L E Dickson, The analytical representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math. 11 (1897),65-120,161-183. [10] M Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970),41-55. [l1J M D Fried, R Guralnick and J Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157-225. [12] R M Guralnick and P Muller, Exceptional polynomials of affine type, J. Algebra, 194 (1997),429-454. [13] R Guralnick and J Saxl, Monodromy groups of polynomials, Groups of Lie Type and their Geometries, Cambridge University Press (1995), 125-150. [14] R Guralnick and M Zieve, Exceptional rational functions of small genus, preprint.

[15] D R Hayes, A geometric approach to permutation polynomials over a finite field, Duke Math. J. 34 (1967), 293-305. [16] H W Lenstra, Jr. and M Zieve, A family of exceptional polynomials in characteristic three, Finite Fields and Applications, London Math. Soc., Lect. Note Ser. 233 (1996), 209-218.

I I

146 [17) R Lidl and H Niederreiter,

Finite Fields, Addison- Wesley, 1983

[18) R Lidl, G L Mullen and G Turnwald,

Dickson Polynomials,

Longman,

1993

[19) MW Liebeck, C Praeger and J Saxl, On the O'Nan-Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. A 44 (1988),389-396. [20) C R MacCluer, On a conjecture of Davenport and exceptional polynomials, Acta Arith. 12 (1967),289-299.

Lewis concerning

[21) P Miiller, New examples of exceptional polynomials, Finite Fields: Theory, Applications and Algorithms, Contemporary Math., Amer. Math. Soc. 168 (1993), 245-250. [22) D Wan, A generalization of the Carlitz conjecture, Finite Fields, Coding Theory and Advances in Communications and Computing, Lect. Notes Pure Appl. Math. (Dekker) 141 (1993),431-432. [23) H Wielandt,

Finite Permutation

Groups, Academic Press, 1964

Algebras and Combinatorics. An International Congress, ICAC'97, Hong Kong. Copyright © 1999 Springer-Verlag Singapore Pte. Ltd. All rights of reproduction in any form reserved.

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