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UNIPOTENT FINITARY LINEAR GROUPS FELIX LEINEN AND ORAZIO PUGLISI 1. Introduction If V is a vector space over the field K, then FGL(F) shall denote the group of all finitary linear transformations of V, that is, the group of all invertible Kendomorphisms g of V for which dim[K,g] is finite. In recent years, finitary linear groups have been a subject of fruitful research (see [2, 3, 4, 8,11,12,14,18]). In the present paper, we shall consider unipotent finitary linear groups. These are either locally finite /^-groups over a field of characteristic p > 0, or torsion-free locally nilpotent groups over a field of characteristic 0 [12, Theorem B]. And by the same theorem, a subgroup G ^ F G L ( F ) is unipotent if and only if it stabilizes a composition series in V. In Section 2, we are concerned with the relationship between unipotent finitary linear groups and the so-called stable matrix groups. Here, a matrix group G is said to be stable, if for every ge G, the matrix g— 1 has onlyfinitelymany non-zero entries. It is fairly easy to show that every countable finitary linear group is stable and that every countable unipotent finitary linear group is unitriangular (Theorem 2.1). It is one of our major results that every countable unipotent finitary linear group actually has a faithful representation, which is unitriangular and stable (Theorem 2.3). In particular, countable unipotent finitary linear groups embed into some of McLain's unitriangular matrix groups (see [15, p. 14]). This also has a consequence for the classification of minimal non-FC groups (Theorem 2.4). Because of Theorem 2.3, it is natural to ask whether certain McLain groups are existentially closed in classes of unipotent finitary linear groups. In order to ensure the existence of existentially closed models, we consider the following classes of objects. For every fixed field K, let 2K be the class of all pairs (G, d), where the group G has a faithful unipotent representation G -> FGL(F) on the AT-space V with local degree function d. Here, d assigns to each finitely generated subgroup of G its degree. The degree of a subgroup .F^ FGL(F) is the minimal dimension of a subspace U of V which contains [V,F] and supplements CV{F) in V. By abuse of notation, we shall often write G in place of (G, d). Within the class fiK, only locally degree-preserving embeddings are considered, that is, embeddings which preserve the degrees of the finitely generated subgroups. If 2p denotes the union of the classes fiK, where /£ ranges over all fields of fixed characteristic p ^ 0, then the ultraproduct argument of [8, Section I] ensures that the class Qp is inductive. Therefore, £ p contains existentially closed pairs. In Section 3, a detailed study of degree-preserving representations of full unitriangular matrix groups offinitedegrees will show that the canonical form of such a representation is not quite natural (Theorem 3.2). As a consequence, McLain groups (with the natural local degree function) cannot be existentially closed in £ p (Theorem 3.3). Received 20 January 1992. 1991 Mathematics Subject Classification 20C20. J. London Math. Soc. (2) 48 (1993) 59-76

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FELIX LEINEN AND ORAZIO PUGLISI

In Section 4 we shall construct an £K-group LK as the union of an ascending chain of full unitriangular groups offinitedegrees with respect to canonical representations. The group LK can be characterized by the fact that it satisfies a fairly strong injectivity property (see Theorems 4.1 and 4.3). If the field Kis algebraically closed or does not contain a proper copy of itself, then LK is, in particular, existentially closed in 2K (Corollary 4.2), and we can embed every £K-group which is the union of an ascending chain of subgroups offinitedegrees into LK (Theorem 4.4). As an abstract group, LK is not a McLain group (Theorem 4.7), although it shares many group theoretical properties with M (Q, K). In Section 5 we shall use Hilbert's Nullstellensatz in order to show that the groups LK over algebraically closed fields K are existentially closed in £ p (Theorem 5.1). The last Section is then devoted to the study of a related class of objects, namely the class 3 p of all pairs (P, d), where P is a transitive/?-group offinitarypermutations with local degree function d, which assigns to every finite subgroup of P the cardinality of its support with respect to a fixed permutation representation of P. Again, embeddings within 3 P have to respect the local degree functions. However, in the permutation group case, the local degree functions make embeddings so rigid that P. Hall's iterated wreath product Wr C p in its natural representation (see [5]) turns out to be the unique existentially closed 3p-group. During this research work, both authors were visitors at Michigan State University, East Lansing, Michigan, USA, supported by a grant of the DFG, respectively CNR. The authors would like to thank J. I. Hall, U. Meierfrankenfeld and R. E. Phillips for many stimulating discussions. 2. Stable form The group of all non-singular and stable matrices over thefieldK, whose rows and columns carry the order type A, will be denoted by GLW(A, K). The corresponding McLain group M{K,K) is the group of all upper unitriangular matrices in GLJA.K). If G ^ FGL(K), then we say that the action ofG is concentrated on U, whenever U is a subspace of V of minimal dimension containing [V, G] and supplementing CV{G). THEOREM 2.1. Suppose that the subgroup G of FGL(F) is the union of an ascending chain of subgroups offinitedegrees. Then there exists a G-invariant subspace U of V with dim U ^ Ko such that the restriction to U is a locally degree-preserving embedding of G into FGL (V) and such that the following hold. (a) A suitable choice of basis yields a stable representation ofG on U. In particular, there exists a locally degree-preserving embedding of G into GLJco, K). (b) If G is unipotent, then a suitable choice of basis yields a unitriangular representation of G on U.

Proof. Let {Gt | ieco} be an ascending chain of subgroups of G of finite degrees with union G. Let U be the union of an ascending chain {Ut\ieco} of finitedimensional subspaces in V with [V, G{] < Ui and V = C/j + C^GJ for all /. Since Ui+1 = Ui + Cv ((?,), we can successively extend a basis B{ of C/( to a basis Bi+1 of Ui+1 in such a way that Bi+1—Bt £ Cv (Gt). It follows that G becomes a stable matrix group with respect to the basis B = (Jiecoî °f ^-

UNIPOTENT FINITARY LINEAR GROUPS

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Let us suppose now that G is unipotent. Then G stabilizes a composition series SP with factors MiINi (jeJ) in U. For every j , choose iel minimal with respect to N, n Ui # Mi n Ut, and pick v]e{MjOUi)—N]. Then {y^êC/J is a basis of £/„ whence {v} \jeJ) is a basis of U which contains precisely one vector out of each jump of the series £f. It follows that with respect to this basis, G is represented as a unitriangular group of matrices, whose rows and columns carry the order type /. At first glance it seems that the methods of proof for the two parts of Theorem 2.1 cannot be combined in order to show that every countable unipotent finitary linear group has a unitriangular and stable representation. Indeed, for every field K, we can give the following example of a countable unipotent group G of finitary linear transformations of a A!-space V, whose given representation on any G-subspace of V cannot be turned into a faithful unitriangular and stable matrix representation of G by a suitable choice of basis. Let V be the .K-space with basis B = {VX\XGA}, where A is the inverse order on the ordinal co+\, and denote the prime field of K by Ko. Regard the group G = ^ FGL(K), where M corresponds to the full McLain group M{A,K0) via matrix representation with respect to B, and where g is given by [va, g] = 0 and \px, g] = vw for all X > co. Then M < G ensures that V has a unique G-composition series 9*, namely the series with factors MJNX {XeA), where Mx = (v^\n < X) and Nx = (y^\n < Xs). In particular, G does not act faithfully on any proper subspace of V. Assume that there exists a basis B of V such that, with respect to B, the matrix representation of G has unitriangular and stable form. Then B must also contain precisely one vector from each jump of the series Sf. Since g is stable with respect to B, we obtain Nx < Cv{g) for some X # co, a contradiction. However, the situation is not completely hopeless. Extend V to a ^T-space V* by introducing an additional basis vector w, which has to be located just above vm. Extend the action of G on V to an action of G on V* via [w, M] = 0 and [w, g) = vw. Then the matrix representation of G with respect to the basis {va, w} U {vk — w \ X > co} is of unitriangular and stable form. The following lemma shows that the above kind of extension is even locally degree-preserving. LEMMA 2.2. Let V ^ V* be vector spaces over the same field and suppose that G < FGL (V). If the action ofG on V extends to an action ofG on V* in such a way that V* = V+CVi,{F)for every {finitely generated) subgroup F ^ G, then this extension is {locally) degree-preserving.

Proof. By hypothesis, [V*,F] = [V,F] and V* = V+CV.{F) for any (finitely generated) subgroup F of G. It follows that V*/{[V*,F] + CV.{F)) = {V+CV.{F))/{[V,F] + CV.{ = VIVfl {[V,F] + CV.{F)) = V/{[V,F) + CV{F)). Hence dv.(F) = dim[V*,F] + codim{[V*,F] + CV.{F)) = dim [V,F] + codim ([ V, F] + CV{F)) = dv{F). By refining the method, which was used to fix the above example, we obtain the following result.

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THEOREM 2.3. If the unipotent subgroup G ofFGL (V) is the union of an ascending chain of subgroups of finite degrees, then G has a locally degree-preserving unitriangular and stable representation. In particular, there exists a locally degree-preserving embedding of G into M (*,X,v>

(3)

fc.v

whenever veB—B0 has the representation v = J^ktVxk ^ k v modulo Cx for suitable xfcveAT. From our choice of the Xik, this transformation does not affect the unitriangular form of the matrix representation of G. Choose the finite subset Bx £ B* such that BQ c B1 and [V*, GJ < . Because of the basis transformation (3), we have B*—Bx £ CK> a n d t n a t [Ti»Ta] = 1- B u t now, [K,[T 1 5 T 2 ],T 3 ] = 0 = [K,T 3 5 [T 1 5 T 2 ]], and the three subgroup lemma yields [T 15 T 2 , T3] = 1, a contradiction. Hence, vx -< v2 -< v3. An iteration of the above argument eventually yields vt -< ... ^< yn_x and [vj+1, r}] # 0 for By replacing vi+1 by a suitable scalar multiple of itself, we may even assume such that [w,Tn_x] = vn_v For that [v)+1,Zj[ = Vj for all j^n — 2. Choose w^v^ 1, choose successively x,e K such that n — 2^j"^ Li

L

k-j+l

k fc+l> i

^ /»

J

and put

Then yn >- vn_x and [y n ,TJ = SnJ+1 • vj for all j ^ n-1. Clearly, B = {v1} ...,vn} is a basis of V. Choose he{2,...,«— 1} minimal with respect to [y,,^] = Sij+1v} for all i^ k+l and ally < «— 1. Assume that k > 2. Then there existsy < k— 1 such that

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[vk, TJ ¥=• 0. But now a straightforward calculation yields the contradiction [T,, rfc] / 1 • This shows that the subgroup (T}\ 1 of M(«, Aô) is represented naturally with respect to the basis B. -ty>. Suppose that this is not the case. Let us observe next that [u((,0,] Then there exists i>j+\ such that u,£ CF( y). As 0, has degree two, its centralizer ,) for some xeK—{0}. in V has codimension one, and so x-vt —vvi+1 i eCy t) ^ But this is clearly a contradiction. For j^n — 2 we now consider the subgroup , whose action is concentrated on the space with basis B} = {vjf vf+1, vj+2}. As matrix groups with respect to the basis Bp we have 0

e,s

I 0 0

o , e,+1sfo 0 0 1

0 0 1

and /I 0 * [0,,0,+1] K and y: [0 P 0, +1 ] -> AT via projection of these groups onto the *-entry of the corresponding matrices. Since we have Ima = Im/? = Imy = Ko )\ — LîJ L}\A)\

— Ui,)+\

A

U

j-

THEOREM 3.2. Let the unipotent finitary linear group G over the field K be the union of an ascending chain of subgroups of finite degrees, and suppose that there is a degree-preserving isomorphism M (n, Ko) = F^G.Ifn^3, then K contains a copy ofK0, and there exist a K-space V with basis B, and a locally degree-preserving embedding of G into FGL(F), such that G is unitriangular and stable with respect to B, and such that F is represented in canonical form with respect to (B-Cv(F))[)(B0[V,F\). JLM48

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Proof. From Theorem 2.3 we may assume without loss, that G = M (Q, K) in its natural representation with respect to a suitable basis B = {vq\qeQ} of the corresponding AT-space V. Denote by Sf the unique G-composition series in V with factors MJNq (qeQ). For v, we V, we shall again write v -< w, whenever the jump in 9* containing v lies below the jump in Sf containing w. The action of Fis concentrated on a «-dimensional subspace U of V. Now Lemma 3.1 yields that AT contains a copy of Ko, and that F acts naturally or conaturally on U with respect to a suitable basis Bo of U. Denote the generators of Fby z^x) as at the beginning of Section 3. Clearly, Bo contains vectors from n different jumps Mq}—Nq (1 ^j^n) of S?. Replacement of the vectors in B which belong to these jumps by tne vectors from Bo yields a basis transformation, which does not affect the unitriangular and stable form of G. Thus, we may assume that Bo c B. Now Facts naturally on U if and only if [V, TJ < [V, zj+1] for 1 ^ y ^ n — 1, while F acts conaturally on U if and only if [K,ii+1] -< [V,r}] for 1 ^ y ^ n — 1. As described in [16, p. 270], an order-reversing permutation of Q gives rise to a locally degree-preserving automorphism a of G such that, if F acts conaturally on U, then [Fjiâ]^ [F,Ti+1a] for l^j^n — 1. We may therefore assume without loss, that F acts naturally on U. Since ,

^

,

, =V

gj+i

for 1 ^j^

n-\,

and since

dim(K/Q(F)) = dimÛ/C^F)) = n-\, the series Sf induces a series y + C?(F) with one-dimensional factors between C$(F) and K. Moreover, {vQ] + Cy(F)\2 ^j ^ «} is a basis of V/Cy(F). Choose r,eQ minimal with respect to Mq + C?(F) < Mr + C${F). Clearly, r} ^ qj+1, and there exists x}eK— {0} such that ^•i; f -p ( | + i eM ! | +C p -(F). If r}^q)+1, then we may replace vT] eBby vf+1 = x} • vfj. Otherwise, we introduce a new basis vector vf+l just below » , and define the action of G on v*+1 via yj^x—y9 +i e C(G). By Lemma 2.2, this does not affect the degree function of G. Now the desired basis B can be obtained from the basis transformation BU{v*,...,v*}^B, which is given by

i-2

whenever veB—B0 has the representation y = X!"_2^^ modulo C^Ci7) for suitable

It is an immediate consequence of Theorem 3.2, that McLain groups cannot be existentially closed in fip, since they are not injective with respect to canonical embeddings of full upper unitriangular matrix groups of finite degree. THEOREM 3.3. A McLain group over the field K (with its natural local degree function) cannot be existentially closed in 2K.

Proof Consider M — M (A, K) in its natural representation on the AT-space V with basis B. Assume, that M is existentially closed in 2K. Then A must be infinite. In particular, if Ko denotes the prime field of K, the group M contains Af3 = M (3, / Q in its. natural representation on a subset {vx, vz, u j of B. Extend B to the basis of a


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X-space V* ^ F b y adding a new basis vector v2, which has to be squeezed into B just below vz. By Lemma 2.2, the action of M on K extends degree-preservingly to an action of M on V* via t>2 — V3ECV*(M). Consider M 4 = M(4, Ao) =

, K) are not isomorphic as £K-groups. On the other hand, they are contained in each other, and hence they have a very similar structure as abstract groups. Nevertheless, it will turn out now that they are even not isomorphic as abstract groups. In order to prove this result, we must study the maximal abelian normal subgroups of LK. Let A be any totally ordered set with Dedekind completion A. Consider the A^space Wwith basis {wA|/leA}, and put : s U > foralUeA, = (wv\ veA,v < A> foralUeA. v

Consider the group UFGL (A, K) of all finitary linear transformations of W, which stabilize the series with factors Wm/Wa) (AeA) in W. For each XeA, respectively AeA, the stabilizer in UFGL (A, K) of the chain 0 < Wa) < W, respectively 0 < Wa > < W is an abelian normal subgroup Ax, respectively Ar, of UFGL (A, K). LEMMA 4.5. The maximal abelian normal subgroups of LK ^ UFGL(/,/Q are precisely the intersections A^Lx (jie T).

Proof. In order to show that Aft(]LK (jneT) is maximal among the abelian subgroups of LK, put Mn = M (3n, K), and consider a matrix geMn—AM. Then gijÔ for some indices / /x. In the case wheny ^ fi, there exist m ^ n and / e / m such that /—n > 1/(2- 3m), and g does not commute with the elementary transvection reMm n A^ given by [vn,z] = 3nyv,. In the case when / > fi, there exist m ^ n and kelm such that k ^ n and i—/i> 1/(2-3m) and g does not commute with the elementary transvection xEMmf]Afi, given by [vn,T] = Snt-vk. In both cases, )) is trivial. The most striking difference between Aut (M(Q,GF (/?))) and Aut(LK) seems to be the non-existence of the inverting automorphism in Aut(LK); since the matrices of LK with respect to the basis {vt \iel} have finite columns and infinite rows, no orderreversing permutation of / can give rise to an automorphism of LK. This fact is also reflected in the structure of the maximal abelian normal subgroups, since LK has no maximal abelian normal subgroup of the form A^- n LK (jiel).

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The question, which diagonal automorphisms and which order automorphisms of UFGL(K) give rise to an automorphism of LK, is more delicate. For example, conjugation by the diagonal matrix with entries aiit = i+\ (iel) maps the elementary transvection reM(3,K), given by [yi, T] = v0 and [vQ,x] = 0, onto a transformation f e U F G L ( F ) , which satisfies where the sequence (in)ne(0 is defined recursively via J'O = | and in = in_i —(§)n+1; and clearly, r$LK, because every matrix in LK has only finitely many distinct entries. However, every diagonal automorphism of M (3 n , A!) can be extended in several ways to a diagonal automorphism of M ( 3 n + \ K), and a successive application yields the maximal number \K\H° of diagonal automorphisms of LK. Correspondingly, an order-preserving permutation n of /, which does not give rise to an order automorphism of LK, can be constructed as follows. Let O'Jneo b e t n e a bove sequence, and define the sequence O n ) n6w recursively via jo = l and ; 3 n + v = y 3 n - v / 3 n + 2 for v = 1,2,3. Note, that inf{/„ | n e co) = J, while inf{jn | neco} = — \. Define n such that in = i for |i| ^ \, and such that n maps each interval [in+1,in] scale-preservingly onto \Jn+1,jn], and correspondingly [-jn, -jn+l] onto [—in, — in+1]. Let f e U F G L ( F ) be the image of the above elementary transformation xeM(3,K) under the order automorphism of UFGL(K), which is induced from n. Assume that f e M (3 m , K) ^ LK for some m. Sincej3m is the smallest index in Im with [v} , f] / 0, the form of the embeddings a n ensures that [vt, f] = 0 for all i e / n ]y8m+3, K/sm+s +hm)l But this contradicts the fact that [v4 1

, f] # 0. Thus n does not give rise to an order

hm+2

J

°

automorphism of LK. However, there still exist enough order automorphisms to make LK characteristically simple. For every n ^ 1, let /in = [3 n /2]. Define intervals for all keZ with — jin ^ k ^ fin — 1, and put Sntk = Jn+i,-Mn+) A.* = -M(3 n + m ,K) by copying matrices. This embedding maps zx,...,zs onto a solution of $ U J in M (3 n+ro , K) ^ LK. It remains to extend the above method to the degree relations 2). To this end, we shall add to 9" a finite set of equations and inequalities over the field K, which will serve as a substitute for 2. Note first that the above conversion of certain ^-matrices A into suitable X-matrices A is not restricted to unitriangular matrices. Moreover, if

74


A is non-singular, then we may add to J' the inequality det^4 # 0 in entries of A, and the conversion will lead to a non-singular matrix A. Since we are dealing throughout with just finitely many matrices, we may assume that non-singular ^-matrices are converted into non-singular AT-matrices. Now, let F be a finitely generated subgroup of G, with a fixed set of generators {h1,..:,ht}^{g1,...,gt}. For all /, let Nt = (hv...,/i,> and Nt = and put / = Nt. Consider a basis B2 of W, which contains a basis of [W,F]. Choose such that A21hlA2 is the matrix with respect to B2, which A2eGL(BltK) corresponds to ht. Then each of the matrices A2'xhlA2 — \ has non-zero entries only in rows corresponding to B2f][W,F]. The same must hold for each A21hlA2 — \, whence dim[W,F]^dim[W,F]. On the other hand, the column vectors of the matrices A21hlA2 — l generate [WtF\. Therefore we can fix a maximum number of linearly independent columns from these matrices. These form a new matrix of rank dim[W,.F] and deletion of some fixed rows leads to a square matrix with non-zero determinant. By adding to J' the inequality that the determinant of this squarematrix is non-zero, we can ensure that dim[J^,/] = dim[W,.F]. This shows, that the dimension [W,F] can be controlled during the conversion. Suppose now by induction that, for some Xe{\,...,t}, the dimension of Q = ^[W,F](Â-I) c a n be controlled during the conversion. Consider a basis Bx+2 of W, which contains bases of [Cx,/ij, of Cx, and of [W,F]. Choose Ak+2eGL(B1} K) such that A\~l2hlAx+2 is the matrix with respect to Bx+2, which corresponds to hv For 1 ^ / < X— 1 each of the matrices Ax~l2hlAx+2 — 1 has zero-columns corresponding to Bx+2 n Cx. Therefore the iiT-span Cx of Bx+2 0 Cx is contained in C^N^. So our inductive hypothesis ensures that Cx = C^ F^X-I)- NOW an application of the above arguments to [Cx, /ij in place of [ W, F] yields that dim [Cx, hx] can be controlled during the conversion. Because Cx+1 is the kernel of the map hx—\:Cx-> [C A ,/JJ, it follows that dimCx+1 = dimCx — dim[Cx,hx] can be controlled during the conversion. The inductionfinallyyields control about dim C[WiF](F). The same argument with Win place of [W,F] yields control about dimCW(F). Therefore, the degree d(F) = dim W- dim CW(F) +dim C[WF](F) can be controlled during the conversion. 6. Finitary permutation groups In this section, we shall determine the existentially closed 3p-groups. Since minimal blocks of 3p-groups are of order/?, it follows immediately from [13, p. 16], that the group Wr C£ is universal in the class 3 p . LEMMA

6.1. The group WrC£ contains a copy of every 3p-group.

In the following Sn will always denote a Sylow/?-subgroup of the symmetric group on pn letters in its canonical wreath product representation of degree pn. LEMMA 6.2. Any locally degree-preserving isomorphism between two copies of Sn in WrCp is induced by an automorphism o

Proof. Put W = Wr C p . For every initial segment n ofco, there is an isomorphism W s Sn wr Wn, where Wn £ Wr C£~n. We regard Sn as the 1-component in this wreath


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product decomposition. Now, let a: Sn -* W be a degree-preserving embedding. Choose meco minimal with respect to Sna? ^ Sm for suitable geW, and assume that n