On the generalized Fitting group of locally finite, finitary groups

U. Meierfrankenfeld

Abstract

Let G be a locally finite, finitary group and F^{∗}(G) the group generated by the Hirsch-Plotkin radical of G and the components of G. Our main theorem asserts that C_{G}(F^{∗}(G)) ≤ F^{∗}(G).

1 Introduction

The main purpose of this paper is to extend the concept of the generalized Fitting group from finite groups to locally finite, finitary groups. Recall that a group G is called locally finite if every finite subset of G lies in a finite subgroup. G is called finitary if there exist a field K and a faithful KG-module V so that [V, g] is finite dimensional for all g ∈ G. G is quasi-simple if it is perfect, and G/Z(G) is simple. A component of G is a non-trivial quasi-simple, subnormal subgroup of G. The layer E(G) is defined as the group generated by the components of G. It is easy to see that distinct components of a group commute. Thus E(G)/Z(E(G)) is semisimple, that is the ( restricted) direct product of simple groups.

Assume for the moment that G is finite. Then the Fitting group F (G) is the largest nilpotent normal subgroup of G and the generalized Fitting group F^{∗}(G) is defined as F (G)E(G). A well known theorem ( see for example [As] ) asserts that C_{G}(F^{∗}(G)) ≤ F^{∗}(G). This means that the group G/Z(F (G)) is a subgroup of the automorphism of F^{∗}(G). Thus a finite group G can be well described in terms of F^{∗}(G). As the strucure of F^{∗}(G) is fairly simple compared with the structure of an arbitray finite group, this explains the importance of the generalized Fitting group in the theory of finite groups.

An infinite group G does not necessarily have a largest nilpotent normal subgroup, but it does have a unique maximal locally nilpotent, normal subgroup, the Hirsch-Plotkin radical (see [Ro, page 58]. We denote the Hirsch Plotkin radical by LN (G) and define the generalized Fitting group F^{∗}(G) of G to be the product LS(G)E(G). The reader should notice that the generalized Fitting group of a group can be trivial. Indeed this is the case for free groups of rank at least two and even for some infinite, locally finite, groups. Surprisingly the situation is much better for locally finite, finitary groups. Indeed, we will prove:

Theorem A Let G be a locally finite, finitary group. Then C_{G}(F^{∗}(G)) ≤ F^{∗}(G).

Similar but different results have been obtained independently by V.V Belyeav [Be, Theorem A] and R.E Phillips [Ph].

As I was kindly reminded by W. Gaschütz, for finite groups the generalized Fitting group can also be described as the set of elements which induce inner automorphism on each chief-factor of the group. 5.11 below implies that same is true for locally finite, finitary groups.

In section 4 we prove that the elements in locally finite, finitary groups have a purely group theoretical property we call bounded. ( See the end of the introduction for the relevant definitions ). In section 5 we prove a structure theorem for arbitray locally finite bounded groups modulo their largest locally solvable normal subgroup. In section 6 we use this structure theorem to prove Theorem A. Finally in section 7 we use the methods and results developed in the earlier sections to show that any locally finite finitary simple group has a Kegel cover which is unipotent by quasi-simple. This last result also has been proved independently in [1, Proposition 4.4] and [Be, Theorem B].

Our results depend on the classification of finite simple groups, but only in form of the Schreier conjecture. That is we assume that the outer automorphism group of every finite simple group is solvable.

We finish the introduction with a list of some of the notation and conventions used throughout this paper.

Let H be group and P a group theoretical property. H is called locally P if every finite subset of H lies in a subgroup of H which property P.

F is a finite group.

G is a locally finite group.

q(F) is the number of non-abelian composition factors in a given composition series for F.

Let X ⊆ Aut(F). Then q(F, X) = ∑_{i=1}^{n} q([F_{i}/F_{i−1}, X]), where

1 = F_{0} < F_{1} < … < F_{n−1} ≤ F_{n} < F

is a maximal X-invariant subnormal series for F.

Let X a finite subset of G and n ≥ 0.

H = {H ≤ G|H finite}.

H(X) = {H ∈ H|X ⊆ H},

X is n-bounded if there exists H_{X} ∈ H(X) such that q(F, X) ≤ n for all F ∈ H(H_{X}) and X is bounded if X is n-bounded for some n.

B_{n} is the set of n-bounded elements, B_{∞} is the set of all bounded elements in G, and G is called bounded if G = B_{∞}.

LS(G) is the subgroup of G generated by all the locally solvable normal subgrous of G. Note that LS(G) itself is locally solvable and so LS(G) is the largest locally solvable normal subgroup of G.

G^{∞} is the group generated by all the perfect finite subgroups of G, i.e G^{∞} is the smallest normal subgroup of G such that G/G^{∞} is locally solvable.

S^{∗}(G) = X^{∞}, where X/LS(G) is group generated by the simple subnormal subgroups of G/LS(G). Note that X/LS(G) is the largest semisimple normal subgroup of G/LS(G).

S_{0}^{∗}(G) = 1 and S_{n+1}^{∗}(G) is defined inductively by S_{n+1}^{∗}(G)/S_{n}^{∗}(G) = S^{∗}(G/S_{n}^{∗}(G)). S_{∞}^{∗}(G) = ⋃_{i=0}^{∞} S_{i}^{∗}(G).

Put S_{∗}^{0}(F) = F^{∞} and inductively for all i > 0 let S_{∗}^{i}(F) be smallest normal subgroup of S_{∗}^{i−1}(F) such that S_{∗}^{i−1}(F)/S_{∗}^{i}(F) is solvable by semisimple. Note that if S_{∗}^{i−1}(F) ≠ 1, then S_{∗}^{i}(F) < S_{∗}^{i−1}(F). Let s(F) be the smallest non-negative integer s such that S_{∗}^{s}(F) = 1. If F is solvable, put S_{∗}(F) = 1, otherwise let S_{∗}(F) = S_{∗}^{s(F)−1}(F).

Note that S_{∗}(F) is a perfect, solvable by semisimple, normal subgroup of F and so S_{∗}(F) ≤ S^{∗}(F). To comprehend the above definitio the reader might verify the following example (the example also shows that S_{∗}(F) is in general a proper subgroup of S^{∗}(F)).

Let F = C_{2} × SL_{2}(5) × Sym(5) ≀ Alt(5). Then

S^{∗}(F) = SL_{2}(5) × Alt(5)^{5}.

F′ = S_{∗}^{1}(F) = S_{2}^{∗}(F) = S_{∞}^{∗}(F) = SL_{2}(5) × Alt(5)^{5}.2^{4}.Alt(5).

S_{∗}^{2}(F) = S_{∗}(F) = Alt(5)^{5}.

s(F) = 2.

Acknowledgement: I would like to thank the Universität Bielefeld and the Universität Kiel for their hospitality.

2 Preliminaries

Lemma 2.1 Let H be a group with H = E(H) and N a normal subgroup of H. Then H = C_{H}(N)N.

Proof: Let L be a component of H. If [N, L] ≤ Z(H), then [N, L, L] = 1 [N, L] = 1 and L ≤ C_{H}(N) If [N, L] ≰ Z(H), then as L/Z(L) is simple, L = [N, L]Z(L) and as L is perfect, L = [N, L] ≤ N. Thus in both cases, L ≤ C_{H}(N)N.

Lemma 2.2 Let N be normal in F and i ≥ 0, then S_{∗}^{i}(F)N/N = S_{∗}^{i}(F/N).

Proof: Clearly S_{∗}^{0}(F)N/N = S_{∗}^{0}(F/N). Suppose i > 0 and put H = S_{∗}^{i−1}(F). By induction HN/N = S_{∗}^{i−1}(F/N). Let P be the inverse image of S_{∗}^{i}(F/N) in F. Then

HN/P ≅ S_{∗}^{i−1}(F/N)/S_{∗}^{i}(F/N)

and so HN/P is solvable by semisimple. Since N ≤ P ≤ HN we have HP = HN. Thus

H/H ∩ P ≅ HP/P = HN/P

and we conclude that H/H ∩ P is semisimple by solvable. Therefore S_{∗}^{i}(F) ≤ H ∩ P ≤ P. On the other hand

HN/S_{∗}^{i}(F)N ≅ H/H ∩ S_{∗}^{i}(F)N = H/(H ∩ N)S_{∗}^{i}(F)

is solvable by semisimple and so P ≤ S_{∗}^{i}(F)N and P = (S_{∗}^{i}(F) ∩ P)N = S_{∗}^{i}(F)N.

Lemma 2.3 Let H ≤ F with S^{∗}(F) ≤ H. Then S^{∗}(H) = S^{∗}(F).

Proof: We assume without loss that LS(F) = 1. Put E = S^{∗}(F). Since LS(F) = 1, E = F^{∗}(F) and E is direct product of the simple subnormal subgroups of F. Since C_{F} (F^{∗}(F)) ≤ F^{∗}(F) ( see [As]) we get C_{F} (E) = 1. Put K = S^{∗}(H) and note that E ≤ K. Since [LS(K), E] ≤ LS(E) = 1 we get LS(K) = 1 and K is semisimple. Thus by 2.1 K = C_{K}(E)E = E.

Lemma 2.4 Let x ∈ K ≤ F, and put s = s(⟨x^{K}⟩). If s ≥ 1 and s(⟨x^{T} ⟩) ≤ s for every subgroup T of F with K ≤ T, then S_{∗}(⟨x^{K}⟩) ≤ S^{∗}(F).

Proof: Put Y = S_{∗}(⟨x^{K}⟩). Since Y is perfect it suffices to show Y ≤ S^{∗}(F)LS(F). Hence we may assume that LS(F) = 1. Put L = ⟨x^{KS^{∗}(F)}⟩ and N = L ∩ S^{∗}(F). Since [x, S^{∗}(F)] ≤ N, L = ⟨x^{K}⟩N. Thus by (2.2), Y N = S_{∗}^{s−1}(L)N. In particular, Y S^{∗}(F) = Y NS^{∗}(F) = S_{∗}^{s−1}(L)S^{∗}(F). Since S_{∗}^{s}(L) = 1, S_{∗}^{s−1}(L) is solvable by semisimple and since S_{∗}^{s−1}(L) and S^{∗}(F) normalize each other we conclude that also Y S^{∗}(F) is solvable by semisimple. Note that both Y and S^{∗}(F) are perfect and so Y S^{∗}(F) = S^{∗}(Y S^{∗}(F)). Furthermore, by 2.3 S^{∗}(Y S^{∗}(F)) = S^{∗}(F) and so Y ≤ S^{∗}(F).

Lemma 2.5 Suppose that G = G^{∞} and G is locally solvable by simple. Then G = ⟨x^{G}⟩ for all x ∈ G \ LS(G).

Proof: As G/LS(G) is simple, G = ⟨x^{G}⟩LS(G). Thus G/⟨x^{G}⟩ is locally solvable and G = G^{∞} ≤ ⟨x^{G}⟩.

Lemma 2.6 Let L be a non-trivial 1-bounded subgroup of G and P a finite, perfect, solvable by simple subgroup of G containing L. Then P is 1-bounded.

Proof: Put H_{P} = ⟨P, H_{L}⟩, let F ∈ H(H_{P}) and F be a maximal P -invariant subnormal series for F. Then L acts non-trivial on at most one of the non-abelian factors of F. By 2.5, P = ⟨L^{P} ⟩ and so the same is true for P. Let W be a non-abelian factor of F not centralized by L. Then clearly L and so also P normalizes the components of W. By maximality of F we conclude that W is simple and so q(F, P) ≤ 1.

Lemma 2.7 Let F be a non-abelian finite group and M a normal subgroup of F minimal by inclusion with respect to [M, F] ≠ 1.

(a) Suppose that C_{F} (M) = Z(F) and Z(F) is cyclic. Then one of the following holds:

(a.1) M = E(M) and F acts transitively on the components of M.

(a.2) M is p-group for some prime p, M = [M, F], F/Z(F)M acts faithfully and irreducibly on M/C_{M}(F). Moreover, M ≅ Ext(p^{1+2m}) or p = 2 and M ≅ C_{4} ∘ Ext(2^{1+2m}). In particular, F/Z(F)M is isomorphic to a subgroup of Sp_{2m}(p).

(b) Suppose that F is a primitive, tensor indecomposable subgroup of GL_{n}(K), where K is an algebraicly closed field. Then (b.a) C_{F} (M) = Z(F) and Z(F) is cyclic. (b.b) (a.1) or (a.2) holds. (b.c) M acts irreducible on K^{n} and in case (a.2) n = p^{m}.

Proof: (a) Since Z(M) ≤ Z(F), Z(M) ≠ M and Z(M) = C_{M}(F) is the unique maximal F invariant subgroup of M. In particular, M/Z(M) is semisimple. If M/Z(M) is not abelian, (1) holds. Thus we may assume that M/Z(M) is an elementary abelian p-group for some prime p. Clearly F acts irreducible on M/Z(M). If M ≠ [M, F], then [M, F] ≤ Z(M) and so M/Z(M) is cyclic and M = Z(M), a contradiction. Thus M = [M, F]. As M has class two and M/Z(M) is elementary abelian, M′ is elementary abelian. As Z(M) is cyclic, this implies |M′| = p. Moreover, M/M′ is elementary abelian and so M = Z(M)E for some E with E ∩ Z(M) = Z(E) = M′ = E′. Thus E is extra-special. If p is odd, M has exponent p and so |Z(M)| = p and E = M. If p = 2 we get that M has exponent four and so either Z(M) ≅ C_{4} or C_{2}. The commutatormap from M/Z(M) × M/Z(M) → M′ defines a non-degenerated symplectic form on M/Z(M). Finally, an automorphism of M which centralizes M/Z(M) and Z(M) is inner and so C_{F} (M/Z(M)) = C_{F} (M)M = Z(F)M and all parts of (a) are established.

(b) Put V = K^{n}. As F is primitive, V is the direct sum of isomorphic irreducible KM-modules. In particular, Z(M) acts by scalars on V, Z(M) ≤ Z(F) and so M is not abelian. Futhermore, if Q does not act irreducible, then N_{GL_{n}(K)}(M) does not act tensor indecomposable on V, a contradiction. Thus M acts irreducible on V and (b.a) holds. In particular, the assumptions of (a) and therefore (a.1) or (a.2) holds. If (a.2) holds, then M = Z(M)E with E extra-special of order p^{1+2m}. As M is irreducible, E is irreducible on V. Thus by [Go, 5.5.5], n = p^{m}.

Lemma 2.8 Let p be a prime and m a postive integer and put n = p^{m}.

(a) 2m < n unless p = 2 and m ≤ 2.

(b) 4m − 2 ≤ n unless p = 2 and m ≤ 3.

(c) n(n − 1)3^{2m−1} ≤ 3^{n−1} unless n = 2, 3, 4 or 8.

(d) n^{2}(n^{2} − 1)3^{4m−1} ≤ 3^{2n−1} unless n = 2, 3, 4 or 8.

Proof: This is readily verified.

Lemma 2.9 (a) Let F be a subgroup of Sp_{4}(2). Then q(F) ≤ 1.

(b) Let F be a subgroup of Sp_{6}(2). Then q(F) ≤ 2.

(c) Let (p, m) ∈ {(2, 1), (2, 2), (2, 3), (3, 1)} and n = p^{m}. Then GL_{m}(p) has no solvable subgroup of order larger than 3^{n−1}/n.

(d) Let (p, m) ∈ {(2, 1), (2, 2), (2, 3), (3, 1)} and n = p^{m}. Then Sp_{2m}(p) has no solvable subgroup of order larger than 3^{2n−1}/n^{2}.

Proof: (a) If q(F) > 1, F has at least two non-abelian composition factors and so

|F | ≤ 60 · 60 = 3, 600 > 720 = |Sp_{4}(2)|.

(b) Note that |Sp_{6}(2)| = 2^{9} · 3^{4} · 5 · 7 = 1, 451, 520 and so |Sp_{6}(2)|/60^{3} < 7. As Sp_{6}(2) has no proper subgroup of index less than 7, q(F) ≤ 2. We remark that, as can be see from the list of maximal subgroups of Sp_{6}(2), q(F) ≤ 1, but we do not need this fact.

(c) If n = 2, 3 or 4, then |GL_{m}(p)| ≤ 3^{n−1}/n. So suppose n = 8. The largest solvable subgroup of GL_{3}(2) has order 24, and 24 ≤ 3^{8−1}/8.

(d) If n = 2 or 3, |Sp_{2m}(p)| ≤ 3^{2n−1}/n^{2}. If n = 4, then 3^{2n−1}/n^{2} = 3^{7}/16 > |Sp_{4}(2)|/6. The only subgroup of Sp_{4}(2) ≅ Sym(6) of index less than 6 are Alt(6) and Sym(6). If n = 8, then 3^{2n−1}/n^{2} > |Sp_{6}(2)|/6. As Sp_{6}(2) is simple, it does not have a proper subgroup of index less then 6.

3 On F^{∗}(G) for locally finite groups

Lemma 3.1 (a) The locally solvable chief-factors of G are elementary abelian p-groups.

(b) Let C be a chief-series for G. Then LN (G) is the largest subgroup of G centralizing all the factors of C.

Proof: [KW, 1.B.4, 1.B.10]

Lemma 3.2 Suppose D is a normal subgroup of G and D is a chief-series for G on D such that D centralizes all the abelian factors of C. Then

(a) [LS(D), D^{∞}] = 1

(b) LS(D) = LN (D).

(c) S^{∗}(D) = E(D)

Proof: Note that by 3.1a the factors of D ∩ LS(D) are abelian and so centralized by D. Let H be a finite subgroup of D and T a finite subgroup of LS(D) and put F = ⟨T^{H}⟩. Then F is finite and H centralizes the factors of D ∩ F. Hence [F, H, i] = 1 for some i ≥ 1.

If H is perfect, we conclude that H centralizes F. Thus (a) holds.

If H = T = F ≤ LS(D), we conclude that H is nilpotent and so (b) holds.

Finally by (a), S^{∗}(D) ∩ LS(D) ≤ Z(S^{∗}(D)) and so (c) holds.

Lemma 3.3 F^{∗}(G) induces inner automorphism on each chief-factor of G.

Proof: Let X/Y be a chief-factor of G. As F^{∗}(G)Y/Y ≤ F^{∗}(G/Y), we may assume that Y = 1.

By 3.1b, [X, LN (G)] = 1 and so we may assume that [X, E(G)] ≠ 1. Then X = [X, E(G)] ≤ E(G) and so by 2.1 E(G) = XC_{E(G)}(X).

Example 3.4 There exists a locally finite group G with G ≠ F^{∗}(G) such that G induces inner automorphism on all of it chief-factors.

Proof: Let H be a finite, perfect, simple group, I an infinite set, H_{i} = H for all ∈ I, E the restricted cartesian product of the H_{i}, i ∈ I, viewed as a normal subgroup of the full cartesian product. Pick 1 ≠ h ∈ H and put h_{i} = h and h^{∗} = (h_{i})_{i∈I}. Finally but G = E⟨h^{∗}⟩. Then it is easy to check that E = F^{∗}(G) and G induces inner automorphism on all of its chief-factors.

4 A bound for q(F, x) in finitary groups

Definition 4.1 (a) Let x ∈ GL_{K}(V). Then deg(x) = deg_{V} (x) = dim[V, x] and

pdeg(x) = pdeg(xZ(GL_{K}(V)) = min_{0≠λ∈K} deg(λ · x).

(b) A PKG-module is a K-vector space V together with a homomorphism

ϕ : G → P GL_{K}(V).

Lemma 4.2 Let K be an algebraicly closed field, x ∈ P GL_{n}(K) and y ∈ P GL_{m}(K). Then pdeg(x⊗y) ≥ m·pdeg(x). In particular, if n, m ≥ 2, then pdeg(x)+pdeg(y) ≤ pdeg(x⊗y).

Proof: Let 0 = V_{0} ≤ V_{1} < V_{2} < … V_{m−1} < V_{m} = V be a maximal chain of y-invariant subspaces in V. As K is algebraicly closed, y acts on each of the V_{i}/V_{i−1} as a scalar. Thus

pdeg(x ⊗ y) ≥ ∑_{i=1}^{m} pdeg_{K^{n}⊗V_{i}/K^{n}⊗V_{i−1}}(x ⊗ y) = ∑_{i=1}^{m} pdeg_{K^{n}}(x) = m · pdeg(x).

To prove the second claim we may assume that pdeg(y) ≤ pdeg(x). Then

pdeg(x ⊗ y) ≥ 2 · pdeg(x) ≥ pdeg(x) + pdeg(y).

Lemma 4.3 (a) If F ≤ P GL_{n}(K), then q(F) ≤ n − 1.

(b) If F ≤ P GL_{n}(K) and x ∈ F, then q(F, x) ≤ 2 · pdeg(x).

Proof: We assume without loss that K is algebraicly closed. Put V = K^{n} and d = pdeg(x). We will prove (a) and (b) by induction on n, and for (b) also by induction on d. Moreover, in (b) we assume without loss that (a) is true and that x is contained in no proper subnormal subgroup of F. In particular, F = ⟨x^{F} ⟩.

Suppose that U is a proper F -submodule in V. Then C_{F} (V/U) ∩ C_{F} (U) is abelian and so

q(F) = q(F/C_{F} (V/U)) + q(C_{F} (V/U)C_{F} (U)/C_{F} (U)) ≤
≤ (dim V/U − 1) + (dim U − 1) = n − 2 < n − 1

Similarly, q(F, x) ≤ 2d.

So we may assume that F acts irreducibly on V. If F is not primitive on V, let Δ be a system of imprimitivity for F on V and put k = |Δ| and m = n/k.

Then q(F) = q(F/C_{F} (Δ))+q(C_{F} (Δ)). Note that F/C_{F} (Δ) is isomorphic to a subgroup of Sym(k) and Sym(k) is isomorphic to a subgroup of GL_{k−1}(K). Thus by induction q(F/C_{F} (Δ)) ≤ k − 2. Moreover, C_{F} (Δ) is contained in the direct product of k subgroups of GL_{m}(K) and hence by induction, q(C_{F} (Δ)) ≤ k(m − 1) and q(F) ≤ (k − 2) + k(m − 1) = km − 2 < n − 1.

Put t = pdeg_{Δ}(x), Γ = C_{Δ}(x), U = ∑ Γ and W = ∑ Δ \ Γ. Let l be the number of non-trivial orbits for x on Δ. As each non-trivial orbit for x in Δ has at least length 2:

(1) 2l ≤ t

Let X_{1}, X_{2}, ...X_{s} be an orbit for x on Γ and X = ∑_{i=1}^{s} X_{i}. Then X = X_{1}[X, x] and so dim[X, x] ≥ (s − 1)m. It follows that pdeg_{W} (x) ≥ (t − l)m. Thus (t − l)m + pdeg_{U}(x) ≤ pdeg_{W} (x) + pdeg_{U}(x) = d and so :

(2) pdeg_{U}(x) ≤ d − tm + lm

Note that the even permutation module for Sym(k) gives rise to a faithful F/C_{F} (Δ)- module Z with dim Z < n and pdeg_{Z}(x) ≤ t − l. Thus by induction :

(3) q(F/C_{F} (Δ), x) ≤ 2(t − l)

Since C_{F} (Δ)C_{F} (W)/C_{F} (W) acts faithfully on W, it is contained in the direct product of at most t copies of GL_{m}(K). Thus (a) implies

(4) q(C_{F} (Δ)C_{F} (W)/C_{F} (W)) ≤ t(m − 1)

Since F = ⟨x^{F} ⟩, x does not fix all the elements of Δ and so U is a proper subspace of V. Futhermore, C_{F} (W) ∩ C_{F} (Δ) acts faithfully on U and so by induction:

(5) q(C_{F} (W) ∩ C_{F} (Δ), x) ≤ 2 · pdeg_{U}(x)

Note that

q(F, x) ≤ q(F/C_{F} (Δ), x) + q(C_{F} (Δ)C_{F} (W)/C_{F} (W)) + q(C_{F} (Δ) ∩ C_{F} (W), x)

and so by (3), (4) and (5):

q(F, x) ≤ 2(t − l) + t(m − 1) + 2 · pdeg_{U}(x)

Hence by (2)

q(F, x) ≤ 2(t−l)+t(m−1)+2d−2tm+2lm = 2d−t(m−1)+2l(m−1) = 2d−(t−2l)(m−1) and so by (1), q(F, x) ≤ 2d.

So we may assume that F acts primitively on V. Suppose that V ≅ U ⊗_{K} W for some at least 2-dimensional PKF -modules U and W. Then

q(F) ≤ q(F/C_{F} (U)) + q(F/C_{F} (W)) ≤ (dim U − 1) + (dim W − 1)) ≤
≤ dim U · dim W − 3 = n − 3 < n − 1.

Moreover, by (4.2) pdeg_{U}(x) + pdeg_{W} (x) ≤ pdeg(x) and so

q(F, x) ≤ q(F/C_{F} (U), x) + q(F/C_{F} (W), x) ≤ 2 · pdeg_{U}(x) + 2 · pdeg_{W} (x) ≤ 2d.

So we may assume that F acts tensor indecomposable on V. Since K is algebraicly closed and F acts primitively on V we conclude that each normal subgroup of F either acts as scalars or acts irreducibly on V. Let N be a normal subgroup of F minimal with respect to N ≰ Z(F). Then N is irreducible and so C_{F} (N) act as scalars on V. In particular, C_{F} (N) = Z(F).

Suppose first that N is not solvable and let Δ be the set of components of N. Then N = ⟨Δ⟩. Put e = |Δ|. Since the outer automorphism group of every finite simple group is solvable, C_{F} (Δ)/N is solvable. Hence q(F) = q(F/C_{F} (Δ)) + q(N) and so

(6) q(F, x) ≤ q(F) = q(F/C_{F} (Δ)) + e.

If e = 1, then F = C_{F} (Δ) and so by (6) q(F) = 1 ≤ n − 1 and q(F, x) ≤ 1 ≤ 2 · pdeg(x). So we may assume that e ≥ 2. As F/C_{F} (Δ) ≤ Sym(Δ) ≤ GL_{e−1}(K) we conclude by induction that q(F/C_{F} (Δ)) ≤ e − 2. Thus by (6):

(7) q(F, x) ≤ q(F) ≤ 2(e − 1) ≤ 2^{e−1}

Since N acts irreducibly on V, V ≅ ⨂_{L∈Δ} V_{L} as a KN-module, where V_{L} is an irreducible PKL-module. In particular, dim V ≥ 2^{e} and so by (7) q(F) ≤ n − 1.

Let L ∈ Δ with [L, x] ≠ 1 and let Λ be the set of all irreducible C_{N}(L)-submodules in V. Then N acts non trivally on Λ. Suppose that x normalizes all elements of Λ. Since L ≤ [L, x], N = [L, x]C_{N}(L) and so N acts trivally on Λ, a contradiction. Pick W ∈ Λ with W ≠ W^{x}. Since C_{N}(LL^{x}) normalizes W and W^{x}, C_{N}(LL^{x}) acts on W/W ∩W^{x}. Therefore dim W/W ∩ W^{x} ≥ 2^{e−2} and so 2^{e−2} ≤ d. Thus by (7) q(F, x) ≤ 2d.

Suppose next that N is solvable. Then by 2.7 Ñ := N/C_{N}(F) is an elementary abelian group of order p^{2m} for some prime p and some positive integer m, N ≅ Ext(p^{1+2m}) or N ≅ C_{4} ∘ Ext(2^{1+2m}), n = p^{m} and C_{F} (Ñ) = Z(F)N is solvable. Also F/C_{F} (Ñ) is isomorphic to a subgroup of Sp_{2m}(p) ≤ GL_{2m}(p). By 2.8a either 2m < p^{m} = n, or p = 2 and m ≥ 2. In the first case by induction, q(F) = q(F/C_{F} (Ñ)) ≤ m − 1 ≤ n − 1. In the second case 2.9a implies q(F) ≤ 1 ≤ n − 1.

Let a ∈ F, then [x, a] = x^{−1}x^{a} has pdeg at most 2d. If [N, x] ≤ Z(F), ⟨x^{F} ⟩ is solvable and q(F, x) = 0. So we may assume that [N, x] ≰ Z(F). Thus N \Z(F) contains an element y with pdeg(y) ≤ 2d. Then 1 ≠ [y, N] ≤ Z(F) and Z(F) contains a non-trivial element z with pdeg(z) ≤ 4d. Since F acts irreducible on V, V = [V, z] and so p^{m} ≤ n ≤ 4d. By 2.8b 2(2m − 1) ≤ p^{m} or p = 2 and m ≤ 3. In the first case by (a)

q(F, x) ≤ q(F) ≤ q(F/C_{F} (Ñ)) ≤ 2m − 1 ≤ p^{m}/2 ≤ 2d.

In the second case, F/C_{F} (Ñ) is isomorphic to a subgroup of Sp_{6}(2) and so by 2.9b q(F) ≤ 2 ≤ 2d. This completes the proof of Lemma 4.3.

5 The structure of B_{∞}

Lemma 5.1 (a) B_{∞} is a normal subgroup of G.

(b) Every finite subgroup of B_{∞} is bounded.

Proof: Let X and Y be n- and m-bounded subsets of G, respectively. Put Z = ⟨X, Y ⟩ and H = ⟨H_{X}, H_{Y} ⟩. Let F ∈ H(H) and 1 = F_{0} ≤ F_{1}... ≤ F_{n−1} ≤ F_{n} = F a chief-series for F. Then [F_{i}/F_{i−1}, Z] = [F_{i}/F_{i−1}, X][F_{i}/F_{i−1}, Y] and so q(F, Z) ≤ q(F, X)+q(F, Y) ≤ n+m. Thus Z is (n + m)-bounded. In particular, any subgroup of G generated by finitely many bounded elements is bounded. Thus (a) and (b) follow.

Lemma 5.2 Let x ∈ B_{n} and K ∈ H(H_{x}). Put F = ⟨x^{K}⟩ and s = s(F).

(a) s ≤ n.

(b) q(F/S_{∗}(F), x) < n

Proof: Put R_{i} = S_{∗}^{i}(F)/S_{∗}^{i+1}(F). As F is normal in K, q(F, x) ≤ q(K, x) ≤ n. Since q(F, x) = ∑_{i=1}^{s} q(R_{i}, x) it suffices to show that q(R_{i}, x) ≥ 1 for all 1 ≤ i ≤ s. Suppose that q(R_{i}, x) = 0 for some 1 ≤ i ≤ s and put \overline{R_{i}} = R_{i}/LS(R_{i}). Then \overline{R_{i}} is a non trivial, perfect, semisimple group with q(\overline{R_{i}}, x) = 0. Thus x centralizes \overline{R_{i}}. As K acts on \overline{R_{i}} and F = ⟨x^{K}⟩ we conclude that F centralizes \overline{R_{i}}. But then \overline{R_{i}} is abelian, a contradiction.

Let B be a normal subgroup of G contained in B_{∞}.

Let x ∈ B. Then by 5.2a there exist d ≥ 0 and K_{x} ∈ H(H_{x}) so that s(⟨x^{K_{x}}⟩) = d and s(⟨x^{K}⟩) ≤ d for all K ∈ H(K_{x}). Let d = d(x) be a minimal such d. Note that x ∈ LS(B) if and only if d(x) = 0.

Let R be the union of the finite and perfect subgroups U of B such that there exists K_{U} ∈ H(U) with U ≤ S^{∗}(F) for all F ∈ H(K_{U}). Note here that R is a characteristcic subgroup of B.

S = ⟨L ≤ B|L is perfect and 1-bounded ⟩.

T = ⟨S_{∗}(⟨x^{K}⟩) | x ∈ B, K ∈ H(K_{x})⟩.

Lemma 5.3 T ≤ R and if B is not locally solvable, R ≠ 1.

Proof: Let x ∈ B and F ∈ H(K_{x}). By choice of d and K_{x} and by 2.4 we have S_{∗}(⟨x^{K}⟩) ≤ S^{∗}(F). Thus S_{∗}(⟨x^{K}⟩) ≤ R and so T ≤ R.

If B is not locally solvable, there indeed exists x ∈ B with d(x) ≠ 0. Hence T ≠ 1 and R ≠ 1.

Proposition 5.4 (a) Let L be perfect and 1-bounded. Then for all F ∈ H(H_{L}), L ≤ S^{∗}(F) and ⟨L^{⟨L^{F} ⟩}⟩ is solvable by simple.

(b) Let L be perfect and 1-bounded. Then ⟨L^{G}⟩ is locally solvable by semisimple and ⟨L^{⟨L^{G}⟩}⟩ is locally solvable by simple.

(c) S ≤ R and S is locally solvable by semisimple.

Proof: (a) Put \overline{F} = F/LS(F) and E = S^{∗}(F). Then C\overline{F} (E) = 1 and so [\overline{E}, \overline{L}] ≠ 1. Since q(F, L) = 1, we conclude that q(F/E, L) = 0 and since L is perfect, L ≤ E. Similarly, [\overline{E}, L] is simple and \overline{L} ≤ [\overline{E}, L]. Hence ⟨\overline{L}^{⟨L^{F} ⟩}⟩ = [\overline{E}, L] and (a) is proved

(b) By (a) Y = ⟨L^{⟨L^{G}⟩}⟩ has a local system of perfect, solvable by simple, finite subgroups. Hence 2.5 implies Y = ⟨y^{Y} ⟩ for all y ∈ Y \ LS(Y) and so Y is locally solvable by simple. Thus ⟨L^{G}⟩ is locally solvable by semisimple.

(c) By (a) L ≤ R and so S ≤ R. By (b), S is locally solvable by semisimple.

Lemma 5.5 S = R, in particular, if B is not locally solvable, S ≠ 1.

Proof: Let U be a perfect finite subgroup of B such that there exists K_{U} ∈ H with U ≤ S^{∗}(F) for all F ∈ H(K_{U}). Let d be minimal with respect to U being d-bounded and put H = ⟨K_{U}, H_{U}⟩. By the minimality of d there exists F ∈ H(H) with q(F, U) = d. Put L = ⟨U^{⟨U^{F} ⟩}⟩. Then clearly q(L, U) = q(F, U) = d. Moreover, since U ≤ S^{∗}(F), it is easy to see that L/LS(L) is the direct product of d simple, perfect subgroups. Let L_{1}, L_{2}, … L_{k} be the non trivial, minimal perfect, normal subgroups of ⟨U^{F} ⟩ and choose notation so that L = L_{1}L_{2} … L_{d}.

We will now prove that for all i, L_{i} is 1-bounded, where we choose H_{L_{i}} = F. Let P ∈ H(F), Q = ⟨U^{P} ⟩ and \overline{P} = P/LS(Q). Since U ≤ S^{∗}(P), \overline{Q} is semisimple. Also L = ⟨L^{U}⟩ ≤ Q. Let \overline{Q}_{1}, \overline{Q}_{2}, …, \overline{Q}_{l} be the components of \overline{Q}. Then \overline{Q} is the direct product of the \overline{Q}_{j}’s. Let \overline{L}_{ij} be the projection of \overline{L}_{i} onto \overline{Q}_{j} and let L_{ij} be the inverse image of \overline{L_{ij}} in P. Let M be the group generated by the L_{ij}’s. Since L_{ij} is a solvable by simple and is normal in M, M is solvable by semisimple. Note that F normalizes M and so q(M, U) ≤ q(F M, U) ≤ d.

Suppose that for some 1 ≤ i ≤ d and some j, [L_{ij}, U] ≤ LS(M). Then L = ⟨U^{L}⟩ and [L_{ij}, L] ≤ LS(M). Hence [\overline{L}_{ij}, \overline{L}_{ij}] is solvable. Since \overline{L}_{ij} is perfect, \overline{L}_{ij} = 1.

Suppose that for some 1 ≤ i < t ≤ d, and some j, L_{ij}LS(M) = L_{tj}LS(M). Since [L_{i}, L_{t}] is solvable we conclude [\overline{L}_{ij}, \overline{L}_{ij}] is solvable and again \overline{L}_{ij} = 1.

For 1 ≤ i ≤ d put p(i) = |\{j | 1 ≤ j ≤ l, \overline{L}_{ij} ≠ 1\}|. Since p(i) ≥ 1 we conclude from the last two paragraphs that

d ≤ q(M, U) ≤ ∑_{i=1}^{d} p(i) ≤ d.

Hence for all 1 ≤ i ≤ d, p(i) = 1 and q(P, L_{i}) = q(Q, L_{i}) = 1. Thus L_{i} is 1-bounded, L ≤ S and since U ≤ L, U ≤ S and R ≤ S. By 5.4(c), S ≤ R and so S = R.

S ≠ 1 follows now from 5.3.

Lemma 5.6 For all n ≥ 0, ⟨B_{n} ∩ B⟩^{∞} ≤ ⟨x ∈ B | d(x) ≤ n⟩^{∞} ≤ S_{n}^{∗}(B). In particular, B^{∞} = S_{∞}^{∗}(B).

Proof: The first containment follows from 5.2(a).

Let x ∈ B with d(x) ≤ n. If n = 0, ⟨x^{G}⟩ is locally solvable and so ⟨x ∈ B | d(x) ≤ 0⟩^{∞} = 1. So suppose that n ≥ 1. By 5.3 and 5.5, T ≤ S. As d(x) ≤ n, S_{∗}^{n−1}(⟨x^{K}⟩) ≤ S_{∗}(⟨x^{K}⟩) ≤ T ≤ S for all T ∈ H(K_{x}). But this implies that in G/S, d(xS) ≤ n − 1. Thus the second containment follows by induction on n.

Theorem 5.7 (a) C_{B}(S/LS(S)) = LS(B).

(b) S = S^{∗}(B).

Proof: We may assume without loss that LS(B) = 1. Then S is semisimple and Z(S) = 1.

Put C = C_{B}(S) and assume that C ≠ 1. Then by 5.5 applied to C instead of B, C has a non-trivial perfect 1-bounded subgroup U. Thus U ≤ S ∩ C = Z(S) = 1, a contradiction, which proves (a).

Since S^{∗}(B) is semisimple, S^{∗}(B) = C_{S^{∗}(B)}(S)S = S.

Proposition 5.8 (a) Let x ∈ B_{n}. Then at most n of the components of S/LS(S) are not centralized by x.

(b) B acts finitarily on the set of components of S/LS(S).

Proof: Without loss LS(B) = 1. Let L be component of B not centralized by ⟨x^{H_{x}}⟩. Since x^{H_{x}} is finite it is easy to see that there exists a non-trivial, finite perfect, N_{H_{x}}(L)- invariant subgroup A_{L} of L with the following property:

(1) Let h ∈ H_{x} such that x^{h} normalizes but does not centralize L. Then [A_{L}, x^{h}] is not solvable.

Choose the A_{L}’s in such a way that A_{L}^{h} = A_{L^{h}} for all h ∈ H_{x}. If x does not normalize L it is easy to see that A_{L} = A′_{L} ≤ [A_{L}, x]. Together with (1) we conclude

(2) If x does not centralize L then [A_{L}, x] ∩ L is not solvable.

Let Δ be finite set of components of B none of which is centralized by x. Put Γ = {L^{h}|L ∈ Δ, h ∈ H_{x}} and A = ⟨A_{L} | L ∈ Γ⟩ = ∏_{L∈Γ} A_{L}. Then A is a finite subgroup of B normalized by H_{x}. By (2), none of groups [A_{L}, x] ∩ L for L ∈ Δ is solvable and thus

|Δ| ≤ q(A, x) ≤ q(AH_{x}, x) ≤ n.

This implies (a). (b) follows immediately from (a).

Lemma 5.9 Let M be a component of B, L ≤ M and F a finite subgroup of N_{G}(M). If L is perfect and 1-bounded, then L is contained in a perfect, 1-bounded, F -invariant subgroup of M.

Proof: Note that M = ⟨L^{⟨L^{M} ⟩}⟩ and so ⟨L^{F} ⟩ ≤ ⟨L^{⟨L^{T} ⟩}⟩ for some T ∈ H. Put K = ⟨F, H_{L}, T ⟩, \overline{K} = K/LS(K) and P = ⟨L^{⟨L^{K}⟩}⟩. Then P is perfect and ⟨L^{F} ⟩ ≤ P. Furthermore, as L is 1-bounded, \overline{P} is a component of \overline{K}. Since F normalizes a non-trivial subgroup of \overline{P}, ( namely \overline{⟨L^{F} ⟩}) we conclude that F normalizes \overline{P} and so also P LS(K). As P is perfect and subnormal in P LS(K), P = (P LS(K))^{∞} and so P is F -invariant. Moreover, by 2.6, P is 1-bounded.

Proposition 5.10 All chief-factors for G on B are semisimple as groups.

Proof: Let Y/X be chief-factor for G on B.

If Y/X is locally solvable, Y/X is semisimple by 3.1(a).

If Y/X is not locally solvable, then LS(Y/X) = 1. Thus by 5.7, Y/X = S^{∗}(Y/X) = E(Y/X) and again Y/X is semisimple.

We are now able to improve 3.2 for bounded groups:

Proposition 5.11 Let C be a chief-series for G on B. Then F^{∗}(B) is the largest subgroup of B inducing inner automorphism on all the factors of C.

Proof: Let D largest subgroup of B inducing inner automorphism on all the factors of C. Since F^{∗}(B) ≤ F^{∗}(G), 3.3implies that F^{∗}(B) ≤ D.

Let D = D ∩ C. Then D also induces inner automorphisms on all the factors of D. In particular D centralizes the abelian factors of D and so by 3.2 LS(D) = LN (D) and S^{∗}(D) = E(D). Let X = C_{D}(E(D). Since S^{∗}(D) = E(D), 5.7 applied to D in place of B, yields X ≤ LS(D). Thus X ≤ LS(D) = LN D ≤ F^{∗}(D).

Let d ∈ D. By 5.8 d centralizes all put finitely many componets of D. As d induces inner autmorphisms on D ∩ E(D), d induces inner automorphism on each component of D. Thus d ∈ XE(B) ≤ F^{∗}(D) ≤ F^{∗}(B). Thus D ≤ F^{∗}(B) and the proposition is proved.

6 Proof of Theorem A

In this section we prove Theorem A. For this let G be a locally finite, finitary group. Then by 4.3, G is bounded. Put B = C_{G}(LN (G)). Then LN (B) = Z(B).

Suppose that LS(B) ≰ Z(B) and let b ∈ LS(B) \ Z(B). Put D = ⟨b^{G}⟩. Then by [MPP, Proposition 1], D is solvable and we can choose E ≤ D with E ≰ Z(B), but E′ ≤ Z(B). Then E′ ≤ Z(E), E is nilpotent and E ≤ LN (B) ≤ Z(B), a contradcition.

Hence LS(B) = Z(B) and so S^{∗}(B) = E(B) ≤ E(G). Thus by 5.7, C_{B}(E(B)) = C_{B}(S^{∗}(B)) ≤ LS(B) ≤ Z(B) and so

C_{G}(F^{∗}(G)) = C_{G}(E(G)LN (G) ≤ C_{B}(E(B)) ≤ Z(B) ≤ F^{∗}(G).

and Theorem A is proved.

7 On Kegel covers for simple, locally finite, finitary groups

In this chapter we investigate Kegel covers of simple, locally finite, finitary groups. Recall that a Kegel covers of the locally finite groups G is a set of pairs {(H_{i}, M_{i}) | i ∈ I} such that

(i) For all i ∈ I, H_{i} is a finite subgroup of G and M_{i} is a maximal normal subgroup of H_{i}.

(ii) For all finite subgroup F of G there exists i ∈ I with F ≤ H_{i} and F ∩ M_{i} = 1.

We remark that every locally finite, simple group has a Kegel cover. See [KW] for a proof.

Lemma 7.1 (a) Let F ≤ Sym(n). Then |LS(F)| ≤ 3^{n−1}.

(b) Let F ≤ GL_{n}(K). Then F has a unipotent by abelian normal subgroup U with |LS(F)/U| ≤ 3^{2n−1}.

(c) Let F ≤ GL_{n}(K) and x ∈ F \ LS(F) with F = ⟨x^{F} ⟩. Put

t = max{|F/LS(F)| · pdeg(x), 4 · pdeg(x)}.

(c.a) The sum of the dimensions of the non-trivial composition factors for F on K^{n} is at most t.

(c.b) F has a unipotent by abelian normal subgroup U with |LS(F)/U| ≤ 3^{2t−1}.

(c.c) Any abelian normal subgroup of F/Unip(F) has rank at most t.

Proof: Note that (c.b) follows from (c.a) and (b). Furthermore any abelian subgroup of GL_{n}(K) has, modulo its unipotent part, rank at most n and so (d.c) follows from (c.a).

(a),(b) and (c.a) are proved simultanously by induction on n. We argue similary as in (4.3). Put V = K^{n}, Ω = {1, 2, …, n}, S = LS(F) and l = |F/S|. For (b) we assume without loss that K is algebraicly closed. As in (4.3) we may assume that G acts transitively on Ω and irrreducibly on V, respectively.

If G acts imprimitively, let Δ be a maximal system of imprimitivity for G on Ω and V, respectively. Put d = |Δ| and k = n/d.

In (a) we get

|S| ≤ |LS(C_{F} (Δ)||LS(F/C_{F} (Δ))| ≤ (3^{k−1})^{d}3^{d−1} = 3^{n−1}.

In (b) let W ∈ Δ. By induction N_{F} (W) has a normal subgroup X containing C_{F} (W) such that X/C_{F} (W) is unipotent by abelian and |LS(N_{F} (W))C_{F} (W)/X| ≤ 3^{2k−1}. Put U = ⋂_{g∈F} X^{g}. Then U is unipotent by abelian and

|LS(F)/U| ≤ |LS(C_{F} (Δ))/U||LS(F/C_{F} (Δ))| ≤ (3^{2k−1})^{d}3^{d−1} = 3^{2n−1}.

In (c) note that k · deg_{Δ}(x) ≤ 2 · pdeg(x). Assume first that S ≰ C_{F} (Δ) and let A be a minimal normal subgroup of SC_{F} (Δ)/C_{F} (Δ). Then A is abelian and either xC_{F} (Δ) ∈ A or [A, x] ≠ 1. In both cases A contains an element 1 ≠ y with deg_{Δ}(y) ≤ 2 · deg_{Δ}(x) ≤ \frac{4}{k} · pdeg(x). Since A acts regularly on Δ we conclude that d ≤ \frac{4}{k} · pdeg(x). Thus n = dk ≤ 4 · pdeg(x) ≤ t. Assume next that S ≤ C_{F} (Δ). Since F acts transitively on Δ, d ≤ |F/C_{F} (Δ)| ≤ |F/S| = l, and we conclude that n ≤ lk ≤ l · pdeg(x) ≤ t.

So we may assume that G acts primitively.

In case (b) suppose that V ≅ X ⊗_{K} Y for some at least 2-dimensional PKF -modules X and Y.Then a finite central extension of G acts faithfully on X ⊕ Y and dim(X ⊕ Y) ≤ n. Thus by the reducible case, (b) holds. So we may assume in case (b) that V is tensorindecomposable.

Let M be normal subgroup of F in S minimal with respect to [M, F] ≠ 1.

In (a), M is an elementary abelian p-group acting regularly on Ω. So n = |M| = p^{m} for some m and |F/M| is isomorphic to a subgroup of GL_{m}(p). Moreover, F/M acts irreducible on M and so has no non-trivial unipotent normal subgroup. Note that every abelian p′ subgroup of GL_{m}(p) has order at most p^{m} − 1 = n − 1. Clearly m < n. Thus by induction and (b), |S/M| ≤ (n − 1)3^{2m−1} and so |S| ≤ n(n − 1)3^{2m−1}. By 2.8c we conclude that either n(n − 1)3^{2m−1} ≤ 3^{n−1} or n ∈ {2, 3, 4, 8}. In the first case |S| ≤ 3^{n−1}. In the second case, 2.9c implies |S/M| ≤ 3^{n−1}/n and again |S| ≤ 3^{n−1}.

In (b) since F is primitive, M is not abelian. Since V is tensor indecomposable and K is algebraicly closed, M acts irreducibly on V. Thus C_{F} (M) = Z(F) is cyclic and we can apply 2.7. We conclude that M ≅ Ext(p^{1+2m}) or M ≅ C_{4} ∘ Ext(2^{1+2m}) for some m that n = p^{m} and that F/MZ(F) acts faithfully M/Z(M). Clearly m < n. As every abelian p′ subgroup of GL_{2m}(p) has order at most p^{2m} − 1 = n^{2} − 1, we get by induction that |S/MZ(F)| ≤ (n^{2} − 1)3^{4m−1}. Thus |S/Z(F)| ≤ n^{2}(n^{2} − 1)3^{4m−1}. By 2.8 either n^{2}(n^{2} − 1)3^{4m−1} ≤ 3^{2n−1} or n ∈ {2, 3, 4, 8}. In the first case |S/Z(F)| ≤ 3^{2n−1}. In the second case 2.9d implies that |S/Z(F)M| ≤ 3^{2n−1}/p^{2m} and so again |S/Z(F)| ≤ 3^{2n−1}.

In (c), M is not abelian and M′ ≤ Z(F). As in 4.3 Z(F) contains a non-trivial element of pdeg at most 4 · pdeg(x) and thus, n ≤ 4 · pdeg(x) ≤ t.

Theorem 7.2 Let G be a simple, locally finite, finitary group and L a perfect 1-bounded subgroup of G. Then there exists a finite subgroup H of G containing L such that ⟨L^{⟨L^{F} ⟩}⟩ is unipotent by quasisimple for all finite subgroups F of G containing H.

Proof: For F ∈ H(H_{L}) define F̃ = ⟨L^{⟨L^{F} ⟩}⟩. For F ∈ H let U(F) be the set of unipotent by abelian normal subgroups of F, u(F) = min{|LS(F)/U||U ∈ U(F)}, U_{+}(F) = {U ∈ U(F)||LS(F)/U| = u(F)}, r(F) = min{rank(U/Unip(U))|U ∈ U_{+}(F)} and U_{∗}(F) = {U ∈ U_{+}(F)|rank(U/Unip(U)) = r(F)}. Replacing L be F̃ for some F ∈ H(H_{L}) we may assume that L is solvable by simple. Let E be a perfect, solvable by simple, finite subgroup of G with E = LS(E)L and let x ∈ L \ LS(L). By 2.5, E = ⟨x^{E}⟩ and so by (7.1), u(E) and r(E) are bounded by a function of |L/LS(L)| and pdeg(x). Hence we can choose E such that first u(E) and then r(E) is maximal. By 2.6, E is 1-bounded. Thus we may and do assume that L = E.

As G has a Kegel cover there exists a subgroup R̂ of G and a maximal normal subgroup M of R̂ so that H_{L} ≤ R̂ and L ∩ M = 1. Put R = ⟨LR̂⟩. As L is perfect and 1-bounded, R is perfect, RM = R̂ and R ∩ M = LS(R). In particular, L ∩ LS(R) = 1 and R is solvable by simple.

Put H = ⟨R, H_{L}⟩ and let F ∈ H(H). To complete the proof of the theorem we will show that F̃ is unipotent by quasi-simple. Note first that F̃ is solvable by simple. As R ≤ F̃ we have L ∩ LS(F̃) ≤ L ∩ LS(R) = 1. Put C = [LS(F̃), L], D = CL and \overline{D} = D/C. Then D = ⟨L^{LS(F̃)}⟩, D is perfect, L ∩ C = 1, \overline{D} ≅ L and LS(D) = LS(L)C. Hence by maximal choice of u(E) (= u(L)), u(D) ≤ u(L). Let U ∈ U_{∗}(D). Then

|LS(D)/U| = u(D) ≤ u(L) = u(\overline{D}) ≤ |LS(\overline{D})/\overline{U}| = |LS(D)/UC| ≤ |LS(D)/U|.

Thus C ≤ U, u(D) = u(L),

|LS(L)/U ∩ L| = |LS(L)U/U| = |LS(D)/U| = u(L)

and U ∩ L ∈ U_{+}(L).

Suppose first that U is unipotent. Then [LS(F̃), L] is a unipotent normal subgroup of LS(F̃) and so [LS(F̃), L] ≤ Unip(F̃). Since F̃ = ⟨LF̃⟩ we conclude [LS(F̃), F̃] ≤ Unip(F̃) and F̃ is unipotent by quasisimple.

Suppose next that U is not unipotent. Then r(D) ≠ 0 and by maximality of r(E) (= r(L)), r(L) ≠ 0. That is, U ∩ L is not unipotent. Let p be a prime and A an elementary abelian p-subgroup of U ∩L with rank(A) = rank(U ∩L/Unip(U ∩L)). Since U ∩L ∈ U_{+}(L), rank(A) ≥ r(L). For X ≤ LLS(F̃), put X^{∗} = XUnip(U)/Unip(U). As [LS(F̃), A, A] ≤ [C, A] ≤ [U, U] ≤ Unip(U) we conclude that [LS(F̃), A]^{∗}A^{∗} = ⟨A^{LS(F̃}⟩^{∗} is an elementary abelian p-group. Since A is a p- and Unip(C) is a p′-group, Unip(U) ∩ AC ≤ C. Hence Unip(U)C ∩A = C ∩A ≤ LS(F̃)∩L = 1. Thus C^{∗} ∩A^{∗} = 1 and in particular [LS(F̃), A]^{∗} ∩ A^{∗} = 1. As rank(U^{∗}) = r(D) ≤ r(L) ≤ rank(A^{∗}), we get [LS(F̃), A]^{∗} = 1 and [LS(F̃), A] ≤ Unip(U). Since F̃ = ⟨AF̃⟩, we conclude [LS(F̃), F̃] ≤ Unip(F̃) and F̃ is unipotent by quasisimple.

Theorem 7.3 Let G be a perfect, simple, locally finite, finitary group.

(a) Let {(H_{i}, M_{i}) | i ∈ I} be a Kegel cover for G with H_{i}/M_{i} perfecr for all i. Let R_{i} the unique minimal normal supplement to M_{i} in H_{i}. Then there exists a finite subgroup H in G so that

{(R_{i}, R_{i} ∩ M_{i}) | i ∈ I, H ≤ H_{i}, H ∩ M_{i} = 1}

is Kegel cover for G and such that all these R_{i}’s are unipotent by quasisimple.

(b) G has a Kegel cover {(H_{i}, M_{i}) | i ∈ I} such that the H_{i}’s are unipotent by quasisimple.

Proof: Clearly (a) implies (b).

To prove (a) note that by 4.3 and 5.5, G has a finite perfect and 1-bounded subgroup L. Let H be as in 7.2. Without loss, H ≤ H_{i} and L ∩ M_{i} = 1 for all i ∈ I. Put H_{i}^{∗} = ⟨L_{i}^{H}⟩. It is easy to see that {(H_{i}^{∗}, H_{i}^{∗} ∩ M_{i}) | i ∈ I} is a Kegel cover for G. By 7.2, H_{i}^{∗} is perfect and unipotent by simple. In particular, H_{i}^{∗} has no proper normal supplement to LS(H_{i}^{∗}). Since H_{i}^{∗}M_{i} = H_{i}, H_{i}^{∗} ∩ M_{i} = LS(H_{i}^{∗}). Thus H_{i}^{∗} = R_{i} and the theorem is proved.

References

[As] M. Aschbacher, Finite Group Theory, Cambridge Unversity Press, Cambridge (1986)

[Be] V.V. Belyaev, Semisimple periodic groups of finitary transformations, Algebra and Logic 32(1)(1983),17-33.

[Go] D. Gorenstein, Finite Groups, Harper, New York (1980)

[1] J.I Hall, Locally finite simple groups of finitary linear transformations, in B. Hartley et al. (eds) Finite and Locally Finite Groups , Kluwer Academic Publishers (1995) 147-188.

[KW] O. Kegel and B.A.F. Wehrfritz, Locally Finite Groups, North Holland (1973)

[MPP] U. Meierfrankenfeld, R.E. Phillips and O. Puglisi, Locally Solvable Finitary Linear Groups, J.London Math.Soc.(2) 47 (1993) 31-40.

[Ph] R.E Phillips, Primitive, locally finite, finitary linear groups ( to appear)

[Ro] D.J.S. Robinson, Finiteness Conditions And Generalized Soluble Groups Part 1, Springer-Verlag, Berlin (1972)

Department of Mathematics Michigan State University East Lansing, Michigam 48824 meier@math.msu.edu