F -stability in finite groups

U. Meierfrankenfeld, B. Stellmacher

May 10, 2006

1 Introduction

Let G be a finite group and p a prime. A subgroup P containing a Sylow p-subgroup of G is a p-parabolic subgroup of G, and P is a local p-parabolic subgroup if in addition O_{p}(P) ≠ 1.

Moreover, G has characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G); and G has parabolic characteristic p if every local p-parabolic subgroup has characteristic p.

The standard examples for groups of parabolic characteristic p are the finite simple groups of Lie type in characteristic p. In these examples every proper parabolic subgroup is a local p-parabolic subgroup, and for maximal parabolic subgroups M the normal subgroup Ω_{1}Z(O_{p}(M)), considered as a GF (p)M-module, has a remarkably restricted structure. In this paper we try to understand this phenomena in arbitrary finite groups.

What kind of properties of the module Ω_{1}Z(O_{p}(M)) should one aim at in general? A possible answer arose during our detailed study of the p-local structure of groups of local characteristic p in [MSS], where a group has local characteristic p if each of its p-local subgroup has characteristic p.

Definition 1.1 Let A be an elementary abelian p-group and V a finite dimensional GF (p)A-module. Then A is

(a) quadratic on V if [V, A, A] = 0,

(b) nearly quadratic on V if [V, A, A, A] = 0 and

[V, A] + C_{V} (A) = [v, A] + C_{V} (A) for every v ∈ V \ [V, A] + C_{V} (A),

(c) an offender on V if |V/C_{V} (A)| ≤ |A/C_{A}(V)|,

(d) a 2F -offender on V if |V/C_{V} (A)| ≤ |A/C_{A}(V)|^{2},

(e) non-trivial on V if [V, A] ≠ 0.

A p-subgroup Y of G is called p-reduced (for G) if Y is elementary abelian and normal in G, and O_{p}(G/C_{G}(Y)) = 1. The largest p-reduced subgroup of G is denoted by Y_{G}; for the existence of Y_{G} see 2.2(a).

Let M be a subgroup of G. Then M is F -stable (in G) if none of the elementary abelian psubgroups of N_{G}(Y_{M})/C_{G}(Y_{M}) are non-trivial offenders on Y_{M}. Similarly, M is 2F -stable (in G) if none of the elementary abelian p-subgroups of N_{G}(Y_{M})/C_{G}(Y_{M}) are non-trivial nearly quadratic 2F -offenders on Y_{M}.

Modules admitting non-trivial 2F -offenders have been investigated by Guralnick, Lawther and Malle in [GLM],[GM1],[GM2], and [L]. They have classified all pairs (V, G), where V is an irreducible GF (p)G-module and G is a known finite almost quasisimple group containing a non-trivial 2F - offender on V.

Their result is a major generalization of earlier results, where G was assumed to contain a nontrivial offender.

For stating our results we need some further definitions.

Definition 1.2 By S(X) we denote the subgroups of G containing X. Let S be Sylow p-subgroup of G.

B(S) := C_{S}(Ω_{1}ZJ(S)),
C^{∗}(G, S) := ⟨C_{G}(Ω_{1}Z(S))), N_{G}(C) | 1 ≠ C char B(S)⟩,

and

C^{∗∗}(G, S) = ⟨N_{G}(J(S)), C_{G}(Ω_{1}Z(S)⟩.

A factorization family for S(S) is a subset F(S) ⊆ S(S) with the following two properties:

(i) For every H ∈ S(S) there exists M ∈ F(S) with H ⊆ C_{G}(Y_{H})M and Y_{H} ≤ Y_{M}.

(ii) If H ∈ S(S) and M ∈ F(S) with M ⊆ C_{G}(Y_{M})H and Y_{M} ≤ Y_{H}, then Y_{M} = Y_{H} and H ≤ M.

Property (i) implies

H/C_{H}(Y_{H}) ≅ HC_{G}(Y_{H})/C_{G}(Y_{H}) ≅ (HC_{G}(Y_{H}) ∩ M)C_{G}(Y_{H})/C_{G}(Y_{H}),

so the action of H on Y_{H} is isomorphic to the action of HC_{G}(Y_{H}) ∩ M on the submodule Y_{H} of Y_{M}. In particular, it suffices to identify M/C_{M}(Y_{M}) and its action on Y_{M} to identify H/C_{H}(Y_{H}) and Y_{H}.

Property (ii) is the crucial one for applications since it has strong consequences. For example, if G is of parabolic characteristic p and S ≤ H ≤ M ∈ F(S) such that M = HC_{M}(Y_{M}), then M is the unique maximal p-local subgroup of G containing H (see 3.5).

Of course, it is not clear a priori that factorization families exist. The existence (and uniqueness) will be established in Theorem 3.4.

Theorem 1.3 Let G be a finite group and S ∈ Syl_{p}(G). There exists a unique factorization family F(S) for S(S) in G. Moreover, at most one member of F(S) is F-stable, and

Ω_{1}Z(S) ≤ Y_{M} and M = N_{G}(Y_{M}) for every M ∈ F(S);

in particular, the elements of F(S) are p-local subgroups of G if S ≠ 1.

In the following results F(S) is always a factorization family for S(S). Recall that a finite group H is p-constrained if H/O_{p′}(H) is of characteristic p.

Theorem 1.4 Let G be a finite group and S ∈ Syl_{p}(G), and let 1 ≠ C char B(S) and M := N_{G}(C). Suppose that there exists N ∈ F(S) that is F-stable.

(a) If C = B(S), then Y_{N} = Y_{M} and N = C_{G}(Y_{M})M = N_{G}(Y_{M}).

(b) If Y_{N} ≤ O_{p}(M), then Y_{M} = Y_{N} and M ≤ N.

(c) If M is p-constrained, then M = O_{p′}(M)(M ∩ N).

Theorem 1.5 Let G be a finite group and S ∈ Syl_{p}(G), and let M ∈ S(S) such that Ω_{1}Z(S) ⊴ M or M = N_{G}(C) for some 1 ≠ C char B(S). Suppose that there exists N ∈ F(S) that is 2F -stable.

(a) If Y_{N} ≤ O_{p}(M), then M ≤ N.

(b) If M is p-constrained, then M = O_{p′}(M)(M ∩ N).

(c) The following hold for any p-constrained H ∈ S(B(S)) with H ⊈ O_{p′}(H)N (where \overline{H} = H/O_{p′}(H)):

(a) \overline{Y_{N}} ≤ O_{p}(\overline{H}).

(b) C_{O_{p}(\overline{H})}(\overline{Y_{N}}) ⊴ \overline{H}.

(c) Y\overline{H} is not F -stable in \overline{H}.

(d) C^{∗∗}(\overline{H}, \overline{T}) ≤ \overline{H ∩ N} < \overline{H}, where B(S) ≤ T ∈ Syl_{p}(H).

For groups of parabolic characteristic p more can be said about the members of the factorization family F(S).

Theorem 1.6 Let G be a finite group of parabolic characteristic p and 1 ≠ S ∈ Syl_{p}(G). Then the members of F(S) are maximal p-local subgroups of G. Moreover, if N ∈ F(S) is 2F -stable and H ∈ S(B(S)) with B(S) ≤ T ∈ Syl_{p}(H), then C^{∗}(H, T) ≤ N.

Corollary 1.7 Let G be a finite group of parabolic characteristic p and S ∈ Syl_{p}(G). If S is contained in at least two maximal p-local subgroups of G, then there exists M ∈ F(S) such that M is not 2F -stable.

Let G and N be as in 1.6, and let H be a p-local subgroup containing S such that H ≰ N. Then by 1.6 C^{∗}(H, S) is a proper subgroup of H. In this case the structure of H can be described precisely using the Local C(G, T)-Theorem proved in [BHS].

The proof of the above theorems relies heavily on two elementary results from [PPS] and [Ste], the L-Lemma and the qrc-Lemma. The authors found it remarkable that these results allow to study finite groups in this context without any K-group assumption.

In fact, using the L-Lemma another result is proved, which is interesting in its own right and which can be used to improve the qrc-Lemma.

Theorem 1.8 Let G be a finite group, S ∈ Syl_{p}(G), and V be a finite dimensional faithful GF (p)Gmodule. Suppose that O_{p}(G) = 1 and S is contained in a unique maximal subgroup of G. Then |A| = |V/C_{V} (A)| for every offender A of G on V.

2 Elementary Properties

In this section G is a finite group, p is a prime, and S ∈ Syl_{p}(G).

Notation 2.1 Let X be a p-subgroup of G. A subgroup P of G is X-minimal if X is contained in a unique maximal subgroup of P and X ≰ O_{p}(P).

Lemma 2.2 Let L be a subgroup of G and P be a p-parabolic subgroup of L.

(a) There exists a unique largest p-reduced subgroup Y_{L} of L.

(b) If Y is a p-reduced subgroup of P with Y ≤ O_{p}(L), then ⟨Y^{L}⟩ is p-reduced for L and so Y ≤ Y_{L}.

(c) If L is of characteristic p, then Y_{P} ≤ Y_{L}.

Proof: (a): Let A and B be p-reduced subgroups of L. It suffices to show that also AB is p-reduced. Then Y_{L} is the product of all p-reduced subgroups of L.

Since A is p-reduced, B ≤ O_{p}(L) ≤ C_{L}(A) and so AB is elementary abelian. Let D be the inverse image of O_{p}(L/C_{L}(AB)). Since C_{L}(AB) ≤ C_{L}(A), DC_{L}(A)/C_{L}(A) ≤ O_{p}(L/C_{L}(A)) and so D ≤ C_{L}(A). By symmetry, D ≤ C_{L}(B) and thus D ≤ C_{L}(A) ∩ C_{L}(B) = C_{L}(AB).

(b): Since P is a p-parabolic subgroup of L, O_{p}(L) ≤ P. Hence [Y, O_{p}(L)] = 1 since Y is p-reduced in P. By assumption Y ≤ O_{p}(L) and so Y ≤ Ω_{1}Z(O_{p}(L)). In particular, V := ⟨Y_{P}^{L}⟩ is an elementary abelian normal subgroup of L.

Since P contains a Sylow p-subgroup of L, there exists S_{0} ≤ P such that S_{0}C_{L}(V)/C_{L}(V) = O_{p}(L/C_{L}(V)) and S_{0} ∈ Syl_{p}(S_{0}C_{L}(V)). As S_{0}C_{P} (V) ⊴ P and C_{P} (V) ≤ C_{P} (Y_{P}), we get that S_{0}C_{P} (Y_{P}) ⊴ P. Hence S_{0}C_{P} (V) ≤ C_{P} (Y_{P}) since Y_{P} is p-reduced in P, and so [V, S_{0}C_{P} (V)] = 1 since S_{0}C_{P} (V) ⊴ P. Thus V is p-reduced for L, and by (a) V ≤ Y_{L}.

(c): As in (b), [Y_{P} , O_{p}(L)] = 1. Since L is characteristic p, Y_{P} ≤ O_{p}(L). So (b) implies Y_{P} ≤ Y_{L}. □

Lemma 2.3 Let X ≤ S ≤ P ≤ G. Suppose that P is X-minimal and N ⊴ P. Then either O^{p}(P) ≤ N and P = XN, or S ∩ N ≤ O_{p}(P). In particular, P = XO^{p}(P) = ⟨X^{P} ⟩.

Proof: Observe that P = NN_{P} (S ∩ N). As P is X-minimal, either NX = P or N_{P} (S ∩ N) = P, and in the second case S ∩ N ≤ O_{p}(P).

Since X ≰ O_{p}(P), S ∩ XO^{p}(P) ≰ O_{p}(P) and so P = XO^{p}(P). A similar argument gives P = ⟨X^{P} ⟩. □

Lemma 2.4 Let A be an F -stable elementary abelian p-subgroup of G, and let Q be a p-subgroup of G with A ⊴ Q. Then the following hold:

(a) A ≤ Z(J(Q)).

(b) ⟨A^{N_{G}(Q)}⟩ is elementary abelian.

Proof: (a): Let B ∈ A(Q). Then B acts on A, and |B| ≥ |C_{B}(A)A| by the maximality of B. Also C_{B}(A) ∩ A ≤ A ∩ B ≤ C_{B}(A) and so C_{B}(A) ∩ A = A ∩ B. Hence

|C_{B}(A)||A||C_{A}(B)|^{−1} ≤ |C_{B}(A)||A||A ∩ B|^{−1} = |C_{B}(A)A| ≤ |B|,

and |A/C_{A}(B)| ≤ |B/C_{B}(A)| follows. The F -stability of A gives [A, B] = 1 and (a) holds. (b): This is a direct consequence of (a) since Z(J(Q)) ⊴ N_{G}(Q). □

Lemma 2.5 Let Q be a normal p-subgroup of G with C_{G}(Q) ≤ Q and Y be an abelian p-subgroup of G. If C_{Q}(Y) ⊴ G and Q normalizes Y, then Y ≤ O_{p}(G).

Proof: Observe that

[Q, Y] ≤ Q ∩ Y ≤ C_{Q}(Y).

Since C_{Q}(Y) ⊴ G this shows that ⟨Y^{G}⟩ centralizes Q/C_{Q}(Y) and C_{Q}(Y). Hence O^{p}(⟨Y^{G}⟩) centralizes Q and since C_{G}(Q) ≤ Q, O^{p}(⟨Y^{G}⟩) = 1 and ⟨Y^{G}⟩ is a p-group. Thus Y ≤ O_{p}(G). □

Lemma 2.6 Let A be a finite elementary abelian p-group and V a finite dimensional GF (p)Amodule. Suppose that A is quadratic on V and [v, A] = [V, A] for every v ∈ V \ C_{V} (A). Then A is a quadratic offender on every A-submodule of V.

Proof: Since every A-submodule of V satisfies the same hypothesis it suffices to show that A is an offender on V. Without loss, [V, A] ≠ 1. Choose W ≤ [V, A] with |[V, A]/W | = p and put \overline{V} = V/W. Let U be the inverse image of C\overline{V} (A) in V. Then [U, A] ≤ W and so [V, A] ≰ [U, A]. Thus U ≤ C_{V} (A) and C\overline{V} (A) = \overline{C_{V} (A)}; in particular, |V/C_{V} (A)| = |\overline{V} /C\overline{V} (A)|. Note that \overline{V} satisfies the hypothesis, so replacing V by \overline{V} we may assume that |[V, A]| = p. Let B < A with |A/B| = p. Since [V, B] is at most 1-dimensional, B in place of A also satisfies the hypothesis of the lemma. Hence by induction on |A|, |V/C_{V} (B)| ≤ |B|.

Let a ∈ A \ B. Since |[V, a]| = p, |V/C_{V} (a)| ≤ p and so also |C_{V} (B)/C_{V} (B) ∩ C_{V} (a)| ≤ p. But C_{V} (A) = C_{V} (B) ∩ C_{V} (a) and so

|V/C_{V} (A)| ≤ |V/C_{V} (B)|p ≤ |B|p = |A|.
□

3 A Partial Ordering

In this section G is a finite group, p is a prime, and S ∈ Syl_{p}(G).

Notation 3.1 Let A and B be subgroups of G. The relation ≪ on the subgroups of G is defined by

A ≪ B : ⟺ A ⊆ C_{G}(Y_{A})B and Y_{A} ≤ Y_{B}.

Furthermore, we define

A^{†} := C_{G}(Y_{A})A and S^{†} := {L ≤ G | L = L^{†}}.

Lemma 3.2 Let L and M be subgroups of G.

(a) Y_{L} ≤ Y_{L^{†}}, L ≪ L^{†}, and (L^{†})^{†} = L^{†}.

(b) S^{†} = {L ≤ G | C_{G}(Y_{L}) ≤ L}.

(c) ≪ is reflexive and transitive.

(d) L ⊆ C_{G}(Y_{L})M if and only if L ≤ C_{G}(Y_{L})N_{M}(Y_{L}).

(e) Suppose that L ⊆ C_{G}(Y_{L})M and L ∩ M is a p-parabolic subgroup of L and M. Then Y_{L} is p-reduced for N_{M}(Y_{L}) and L ≪ N_{M}(Y_{L}).

(f) If L = L^{†}, then L ≪ M if and only if Y_{L} ≤ Y_{M} and L = C_{G}(Y_{L})(L ∩ M).

(g) Restricted to S^{†}, ≪ is a partial ordering.

Proof: (a): Clearly Y_{L} is a p-reduced subgroup of L^{†}, so Y_{L} ≤ Y_{L^{†}}. Thus C_{G}(Y_{L^{†}}) ≤ C_{G}(Y_{L}) ≤ L^{†} and L^{†} = (L^{†})^{†}.

(b): This is an immediate consequence of the definition of L^{†}.

(c): Obviously ≪ is reflexive. If A, B, C ≤ G with A ≪ B and B ≪ C, then Y_{A} ≤ Y_{B} ≤ Y_{C} and so Y_{A} ≤ Y_{C}. Also C_{G}(Y_{B}) ≤ C_{G}(Y_{A}) and hence

A ⊆ C_{G}(Y_{A})B ⊆ C_{G}(Y_{A})C_{G}(Y_{B})C = C_{G}(Y_{A})C.

Thus A ≪ C and ≪ is transitive.

(d): If L ⊆ C_{G}(Y_{L})M then L ≤ N_{G}(Y_{L}) ∩ C_{G}(Y_{L})M = C_{G}(Y_{L})N_{M}(Y_{L}). The other direction is obvious.

(e): Since L ∩ M is a p-parabolic subgroup of L,

Y_{L} ≤ O_{p}(L) ≤ L ∩ M ≤ N_{M}(Y_{L}),

so Y_{L} is an elementary abelian normal subgroup of N_{M}(Y_{L}). Since L ∩ M is a p-parabolic subgroup of M, C_{G}(Y_{L})(L ∩ M) and thus also C_{G}(Y_{L})L are p-parabolic subgroups of C_{G}(Y_{L})N_{M}(Y_{L}).

As Y_{L} is a p-reduced subgroup of C_{G}(Y_{L})L, 2.2(b) shows that Y_{L} = ⟨Y_{L}^{C_{G}(Y_{L})N_{M} (Y_{L})}⟩ is p-reduced for C_{G}(Y_{L})N_{M}(Y_{L}). Hence Y_{L} is also a p-reduced subgroup of N_{M}(Y_{L}). Thus Y_{L} ≤ Y_{N_{M} (Y_{L})} and so L ≪ N_{M}(Y_{L}).

(f): Since L ∈ S^{†} we have C_{G}(Y_{L}) ≤ L and so L ⊆ C_{G}(Y_{L})M implies L = C_{G}(Y_{L})(L ∩ M). Now (f) is obvious.

(g): Let L, M ∈ S^{†} with L ≪ M and M ≪ L. Since Y_{L} ≤ Y_{M} ≤ Y_{L}, we have Y_{L} = Y_{M}. By (f) L = C_{G}(Y_{L})(L ∩ M) and M = C_{G}(Y_{M})(M ∩ L). Hence Y_{M} = Y_{L} gives L = M. So the restriction of ≪ to S^{†} is anti-symmetric. Now (g) follows (c). □

Notation 3.3 Put S^{†}(S) := {L ∈ S^{†} | S ≤ L}. According to 3.2(g) ≪ restricted to S^{†}(S) is a partial ordering on S^{†}(S). We denote the set of maximal elements of S^{†}(S) with respect to ≪ by F(S).

Theorem 3.4 F(S) is the unique factorization family for S(S).

Proof: Let G be a factorization family for S(S) and let M ∈ G. Clearly M ≤ M^{†} and by 3.2(a), Y_{M} ≤ Y_{M^{†}}. So Condition (ii) of 1.2 gives M = M^{†}. Thus M ∈ S^{†}(S) and G ⊆ S^{†}(S).

Now let G be any subset of S^{†}(S). Then Condition (i) of 1.2 is fulfilled for G if and only if for each L ∈ S(S) there exists M ∈ G with L ≪ M. Since L ≪ L^{†} and ≪ is transitive by 3.2, we conclude that G fulfills (i) if and only if G contains all the maximal elements of S^{†}(S) with respect to ≪. And Condition (ii) holds if and only if all elements of G are maximal with respect to ≪ in S^{†}(S). Thus F(S) is the unique factorization family for S(S). □

Lemma 3.5 Let M ∈ F(S) and H ∈ S(S) with M = C_{M}(Y_{M})(M ∩ H). If H is p-constrained, then H = O_{p′}(H)(H ∩ M). In particular, if G is of parabolic characteristic p and S ≤ L ≤ M with M = C_{M}(Y_{M})L, then M is the unique maximal p-local subgroup of G containing L.

Proof: Put \overline{H} = H/O_{p′}(H). Since M = C_{M}(Y_{M})(H ∩M), Y_{M} is p-reduced for H ∩M and \overline{Y_{M}} is a p-reduced subgroup of \overline{H ∩ M}. So by 2.2(c) , \overline{Y_{M}} ≤ Y\overline{H}. Let Y ≤ S with \overline{Y} = Y\overline{H} and K := N_{H}(Y). Then by the Frattini argument, H = O_{p′}(H)K. It follows that Y is a p-reduced subgroup of K, so Y_{M} ≤ Y ≤ Y_{K}.

As Y O_{p′}(H) ∩ M = Y (O_{p′}(H) ∩ M), we also get, using the Frattini argument one more time,

H ∩ M = (O_{p′}(H) ∩ M)(K ∩ M) = O_{p′}(M ∩ H)(K ∩ M).

Thus M = C_{M}(Y_{M})(H ∩ M) ≤ C_{G}(Y_{M})K since O_{p′}(M ∩ H) centralizes Y_{M}. Now 1.2(ii) implies that K ≤ M and so H = O_{p′}(H)(H ∩ M). Hence the first statement holds.

To prove the the second statement, let H be a p-local subgroup containing L. Then M = C_{M}(Y_{M})L implies M = C_{M}(Y_{M})(H ∩ M). On the other hand H is of characteristic p since G has parabolic characteristic p, so H is p-constrained and O_{p′}(H) = 1. Hence by the first statement H = O_{p′}(H)(H ∩ M) ≤ M. □

Lemma 3.6 Let M ∈ F(S), S_{0} := C_{S}(Y_{M}) and M_{0} := N_{M}(S_{0}).

(a) M = C_{M}(Y_{M})M_{0}, S_{0} = O_{p}(M_{0}) and C_{S}(S_{0}) ≤ S_{0}.

(b) Ω_{1}Z(S) ≤ Y_{M} = Y_{M_{0}} = Ω_{1}Z(S_{0}).

Proof: (a): The Frattini argument gives M = C_{M}(Y_{M})M_{0}. Hence O_{p}(M_{0}) = S_{0}, since Y_{M} is p-reduced. Clearly Y_{M} ≤ Ω_{1}Z(S_{0}), and so

C_{S}(S_{0}) ≤ C_{S}(Ω_{1}Z(S_{0})) ≤ C_{S}(Y_{M}) ≤ S_{0}.

(b): Let S_{0} ≤ S_{1} ≤ S with

S_{1}C_{M_{0}}(Ω_{1}Z(S_{0}))/C_{M_{0}}(Ω_{1}Z(S_{0})) = O_{p}(M_{0}/C_{M_{0}}(Ω_{1}Z(S_{0}))).

Then S_{1}C_{M}(Y_{M})/C_{M}(Y_{M}) is a normalized by M_{0}C_{M}(Y_{M}) = M. Since O_{p}(M/C_{M}(Y_{M})) = 1 we get S_{1} ≤ C_{M}(Y_{M}), so S_{1} = S_{0} = O_{p}(M_{0}) by (a), and Ω_{1}Z(S_{0}) is p-reduced for M_{0}. Together with 3.2(a) this gives

Y_{M} ≤ Ω_{1}Z(S_{0}) ≤ Y_{M_{0}} ≤ Y_{M_{0}^{†}}.

In particular M ≪ M_{0}^{†}, and the maximality of M yields Y_{M} = Y_{M_{0}^{†}}. Now (b) follows, since also Ω_{1}Z(S) ≤ Ω_{1}Z(S_{0}). □

Proof of Theorems 1.3 and 1.4:

By 3.4 F(S) is the unique factorization family for S(S). Let M ∈ F(S). By 3.6(b) Ω_{1}Z(S) ≤ Y_{M} and by 3.2(e) M ≪ N_{G}(Y_{M}). Hence the maximality of M gives M = N_{G}(Y_{M}).

Assume that there exists N ∈ F(S) that is F -stable, i.e. Y := Y_{N} is F -stable in N_{G}(Y_{N}) = N. Then by 2.4 B(S) ≤ C_{G}(Y) and

N = C_{G}(Y)N_{N}(B(S)) ⊆ C_{G}(Y)L, where L := N_{G}(B(S);

in particular Y ≤ Ω_{1}Z(B(S)) ≤ O_{p}(N_{G}(B(S))). Now 2.2(b) implies that Y ≤ Y_{L} and so by 1.2(ii) Y = Y_{L}. It follows that N = N_{G}(Y_{L}). In particular, N is the unique F -stable member of F(S). This finishes the proof of 1.3 and also shows 1.4(a).

Now let 1 ≠ C char B(S) and put M := N_{G}(C). Then N_{N}(B(S)) ≤ M and thus also N = C_{G}(Y)(M ∩ N). Suppose that Y_{N} ≤ O_{p}(M). Then as above 2.2(b) implies that Y_{N} ≤ Y_{M}, and by 1.2(ii) Y_{N} = Y_{M} and M ≤ N. So 1.4(b) holds.

Suppose next that M is p-constrained. From N = C_{G}(Y)(N ∩ M) and 3.5 we get that M = O_{p′}(M)(M ∩ N). Hence 1.4(c) holds. □

4 The L-Lemma and the qrc-Lemma

In this chapter we will work with the following hypothesis.

Hypothesis 4.1 Let P be a finite group of characteristic p, T ∈ Syl_{p}(P), Y ⊴ T, and R := C_{T} (Y). Suppose that P is RO_{p}(P)-minimal with M being the unique maximal subgroup of P containing RO_{p}(P).

Notation 4.2 Let X be a finite group and V a finite dimensional GF (p)X-module. By c(V, X) we denote the number of non-central chief factors of X in V (in a given chief series). We define q(V, X) := 0 if every quadratically acting subgroup of X already centralizes V, and

q(V, X) := min{log_{|A/C_{A}(V)|} |V/C_{V} (A)||A ≤ X, [V, A, A] = 1 ≠ [V, A]}

otherwise. Moreover, r(V, X) := 0 if V does not possess non-central X-chief-factors, and

r(V, X) := min{q(C, X) | C non-central X-chief-factor on V }

otherwise.

Lemma 4.3 (L-Lemma) Assume Hypothesis 4.1. Let A be a subgroup of T such that A ≰ O_{p}(P). Then there exists a subgroup L ≤ P with AO_{p}(P) ≤ L satisfying:

(a) AO_{p}(L) is contained in a unique maximal subgroup L_{0} of L, and L_{0} = L ∩ M^{g} for some g ∈ P.

(b) L = ⟨A, A^{x}⟩O_{p}(L) for every x ∈ L \ L_{0}.

(c) L is not contained in any P -conjugate of M.

Proof: See [PPS]. □

The next lemma is very similar to [Ste, 3.3].

Lemma 4.4 Assume Hypothesis 4.1. Suppose V := ⟨Y^{P} ⟩ is elementary abelian, C_{O_{p}(P)}(Y) ⋬ P and c(V, P) = 1. Then [O_{p}(P), O^{p}(P)] is non-trivial quadratic offender on Y.

Proof: Since P is RO_{p}(P)-minimal, we get from 2.3 that P = RO^{p}(P)O_{p}(P). Put

Q := [O_{p}(P), O^{p}(P)], W := [V, O^{p}(P)] and D := C_{V} (O^{p}(P)).

Since c(V, P) = 1, W/W ∩ D is chief-factor for P on V. Hence [W, O_{p}(P)] ≤ D. Note that P = T O^{p}(P) normalizes Y W and so V = Y W. Thus [Y, Q]D ⊴ P. Observe that R centralizes [Y, Q]D/D. Since O^{p}(P) ≤ ⟨R^{P} ⟩ we conclude that [Y, Q, O^{p}(P)] ≤ D. Hence O^{p}(P) centralizes [Y, Q]. So P = T O^{p}(P) normalizes [Y, Q] and [V, Q] = [Y, Q] ≤ D. It follows that

[V, O^{p}(P), O_{p}(P)] = [W, O_{p}(P)] ≤ D and [O_{p}(P), O^{p}(P), V] = [Q, V] ≤ D.

Hence the Three Subgroup Lemma implies [V, O_{p}(P), O^{p}(P)] ≤ D and so

(∗)  [V, O_{p}(P)] ≤ D.

Pick x ∈ Y \ D. Since R centralizes x we conclude from (∗) that P = RO^{p}(P)O_{p}(P) normalizes ⟨x^{O^{p}(P)}⟩D and so W ≤ ⟨x^{O^{p}(P)}⟩D. Put X := [x, Q]. Since X ≤ D it follows that

[W, Q] ≤ [⟨x^{O^{p}(P)}⟩D, Q] = [x, Q] = X.

As [V, Q, O^{p}(P)] = 1 ≤ X and [V, O^{p}(P), Q] = [W, Q] ≤ X, the Three Subgroup Lemma implies [Q, O^{p}(P), V] ≤ X. Since [Q, O^{p}(V)] = Q we get [V, Q] = X. In particular,

[y, Q] = X for every y ∈ Y \ C_{Y} (Q).

Now 2.6 shows that Q is a quadratic offender on Y.

If Q acts trivially on Y, then Q ≤ C_{O_{p}(P)}(Y) and so C_{O_{p}(P)}(Y) ⊴ T O^{p}(P) = P, a contradiction. □

Lemma 4.5 Let L be a finite group acting on a p-group E, and let A and B be p-subgroups of L and X and Z subgroups of E. Suppose that

(i) B ≰ O_{p}(L), (ii) [E, A] ≤ X ≤ C_{E}(A) and [E, B] ≤ Z ≤ C_{E}(B), (iii) L is AO_{p}(L)-minimal and [E, O^{p}(L)] ≠ 1, (iv) X is normalized by E and O_{p}(L), X is abelian, and E = ⟨X^{L}⟩. Then

(a) C_{B}(E) ≤ B ∩ O_{p}(L),

(b) E = X^{g}Z = X^{g}C_{E}(B) for some g ∈ L,

(c) Z = [E, B]C_{Z}(E) = [E, b]C_{Z}(E) for all b ∈ B \ O_{p}(L),

(d) [B, E, E, E] = 1,

(e) |Z/C_{Z}(E)| ≤ |ZD/D| ≤ |E/C_{E}(B)|,

(f) |BO_{p}(L)/O_{p}(L)| ≤ |E/C_{E}(B)|. Proof: By (iii) there exists a unique maximal subgroup L_{0} of L containing AO_{p}(L), and by 2.3 (⋂_{g∈L} L_{0}^{g})/O_{p}(L) is a p′-group.

Pick b ∈ B \ O_{p}(L). Then there exists g ∈ L with b ∉ L_{0}^{g}. Put H := ⟨A^{g}, b⟩. Then L = HO_{p}(L) since H ≰ L_{0}^{g}. Furthermore, we put D := ⋂_{g∈L} X^{g}.

(a): Again by (iii) O^{p}(L) ≰ C_{L}(E), so 2.3 shows that C_{L}(E)/C_{O_{p}(L)}(E) is a p′-group. Now (a) follows.

(b): By (iv) O_{p}(L) normalizes X, so X^{L} = X^{H}. It follows that D = ⋂_{h∈H} X^{h} and C_{X^{g}}(b) ≤ C_{X^{g}}(H) ≤ D. From E = ⟨X^{L}⟩ = ⟨X^{H}⟩ and (ii) we conclude that

(∗)  E = X^{g}[E, H] = X^{g}[E, A^{g}][E, b] = X^{g}[E, b] = X^{g}Z.

Since Z ≤ C_{E}(B) by (ii), (b) holds.

(c): Since X^{g} ∩ Z ≤ C_{X^{g}}(H) ≤ D, we get X^{g} ∩ Z = D ∩ Z and so by (b) and (∗)

|E/X| = |E/X^{g}| = |ZX^{g}/X^{g}| = |Z/Z ∩ X^{g}| = |Z/Z ∩ D|.

Moreover, using (∗)

Z = (X^{g} ∩ Z)[E, b] ≤ (Z ∩ D)[E, b] ≤ C_{Z}(E)[E, b] ≤ C_{Z}(E)[E, B] ≤ Z,

and so (c) holds.

(d): Since L is AO_{p}(L)-minimal, A ≰ O_{p}(L) and so (b) can be applied with A and A^{g} in place of A and B. Then E = XX^{t} for some t ∈ L; in particular [X^{t}, X] ≤ X ∩ X^{t} ≤ D ≤ Z(E). Thus E′ ≤ D ≤ Z(E) and [B, E, E, E] ≤ [E′, E] = 1. So (d) holds.

(e): By (ii) and (b)

C_{E}(B) ≤ C_{X^{g}}(B)Z = C_{X^{g}}(H)Z = DZ.

Hence

|E/C_{E}(B)| ≥ |E/ZD| = |X^{g}ZD/ZD| = |X^{g}/X^{g} ∩ ZD| = |X^{g}/D|.

On the other hand, by (b) |E/X| = |Z/Z ∩ X^{g}| = |Z/Z ∩ D|, while the same result applied to A in place of B gives |E/X| = |X/D| = |X^{g}/D|. Since D ≤ Z(E) this gives

|E/C_{E}(B)| ≥ |Z/Z ∩ D| ≥ |Z/C_{Z}(E)|.

(f): Let x ∈ X^{g} \ D and suppose that [x, b] ∈ D. Then ⟨x⟩D is normalized by ⟨X^{g}, b⟩ = H and so x ∈ D, a contradiction. This shows that [x, c] ∉ D for every c ∈ B \ O_{p}(L). Since B acts quadratically on the abelian group E/D we conclude

|[x, B]D/D| = |{[x, c]D | c ∈ B}| ≥ |BO_{p}(L)/O_{p}(L)|.

Note that by (ii), [x, B]D ≤ ZD and so (f) now follows from (e). □

Theorem 4.6 Assume Hypothesis 4.1. Let V be a finite dimensional GF (p)P -module such that [V, O_{p}(P)] = 0 and [V, O^{p}(P)] ≠ 0. Then q(V, P) = 0 or q(V, P) ≥ 1.

Proof: Let A ≤ T be a quadratic on V with [V, A] ≠ 0. We need to show that that |V/C_{V} (A)| ≥ |A/C_{A}(V). The proof is by induction on |A|.

Let Y be a non-central P -chief factor in V. By 2.3 C_{T} (Y) ≤ O_{p}(P) ≤ C_{T} (V). It follows that

|Y/C_{Y} (A)| ≤ |V/C_{V} (A)| and |A/C_{A}(Y)| = |A/C_{A}(V)|

for every A ≤ T. Hence we may assume that

1^{∘} V is a non-trivial simple P -module.

We now apply 4.3. Then there exists A ≤ L such that L has the properties given in 4.3. In particular, there exists g ∈ P such that A ≤ T^{g} ∩ L ∈ Syl_{p}(L), and L ∩ M^{g} is the unique maximal subgroup of L containing AO_{p}(L). Put U := ⟨C_{V} (T^{g})^{L}⟩. 2^{∘} [U, O^{p}(L)] ≠ 0 and [U, A] ≠ 0.

By (1^{∘}) C_{V} (T^{g}) is not P -invariant, so N_{P} (C_{V} (T^{g})) ≤ M^{g}. Since L ≰ M^{g}, we get that [U, O^{p}(L)] ≠ 0 and thus also [U, A] ≠ 0. 3^{∘} Put D := C_{A}(U). Then |A/D| ≤ |U/C_{U}(A)|.

Observe that by the definition of U, [U, O_{p}(L)] = 0. Thus, for E := U, B := A, and X := Z := C_{U}(A), L satisfies the hypothesis of 4.5. By 4.5(f)

|A/D| = |A/C_{A}(U)| ≤ |A/A ∩ O_{p}(L)| ≤ |U/C_{U}(A)|.

So (3^{∘}) holds. 4^{∘} |D/C_{D}(V)| ≤ |V/C_{V} (D)|.

Since [U, A] ≠ 0, D < A and (4^{∘}) follows by induction on |A|.

Using (3^{∘}) and (4^{∘}) we compute

\begin{aligned}|A/C_{A}(V)| & = |A/D||D/C_{D}(V)| ≤ |U/C_{U}(A)||V/C_{V} (D)| \\ & ≤ |C_{V} (D)/C_{V} (A)||V/C_{V} (D)| = |V/C_{V} (A)|. \end{aligned}

□

The next lemma is a variation of [Ste, 3.2]. Lemma 4.7 (qrc-Lemma) Assume Hypothesis 4.1. Let V := ⟨Y^{P} ⟩. Suppose that

(i) Y ≤ Ω_{1}Z(J(O_{p}(P))), (ii) C_{O_{p}(P)}(Y) ⋬ P, (iii) J(R) ≰ O_{p}(P). Then V ≤ Ω_{1}Z(J(O_{p}(P))), V ≠ Y, N_{L}(Y) ≤ M, [V, O^{p}(P)] ≠ 1, C_{T} (V) ≤ O_{p}(P) and there exists A ∈ A(R) with

[V, A, A] = 1 ≠ [V, A] and A ≰ O_{p}(P).

Moreover, one of the following holds, where q := q(Y, O_{p}(P)), r := r(V, P) and c := c(V, P):

(a) 0 ≠ q ≤ 1.

(b) 2 ≤ c, 1 ≤ r, and (q − 1)(rc − 1) ≤ 1. In particular, 0 ≠ q ≤ 2.

Proof: By (i) V ≤ ZJ(O_{p}(P)). If V = Y, then C_{O_{p}(P)}(Y) ⊴ P, a contradiction to (ii). Hence V ≠ Y and RO_{p}(P) ≤ T ≤ N_{P} (Y) < P. Since P is RO_{p}(P)-minimal we conclude that N_{P} (Y) ≤ M. So [V, O^{p}(P)] ≠ 1. Hence 2.3 gives C_{T} (V) ≤ O_{p}(P). In particular, (P, Y) satisfies Hypothesis III of [Ste].

As C_{T} (V) ≤ O_{p}(P), (iii) shows that there exists A ∈ A(R) such that [V, A] ≠ 1. By the Timmesfeld Replacement Theorem [KS] we may assume that [V, A, A] = 1. Moreover, (i) implies that A ≰ O_{p}(P).

Suppose that c = 1. Then 4.4 shows that (a) holds. Thus, we may assume from now on that c ≥ 2.

Suppose that [A∩O_{p}(P), V] = 1. Again by 2.3 O_{p}(P)∩A = C_{A}(V) = C_{A}(U) for every non-central P -chief factor U of V. On the other and, by the maximality of A, |V/C_{V} (A)| ≤ |A/C_{A}(V)| and thus also |U/C_{U}(A)| ≤ |A/C_{A}(U)|. Hence 4.6 implies that c = 1, which contradicts our assumption. We have shown that [A ∩ O_{p}(P), V] ≠ 1; in particular q ≠ 0. Now [Ste, 3.2 (c)] and 4.6 yield (b). □

Lemma 4.8 Assume hypothesis 4.1. Suppose that

(i) C_{O_{p}(P)}(Y) ⋬ P,

(ii) Y ≤ Z(J(O_{p}(P))) ∩ Z(J(T)),

(iii) J(T) ≰ O_{p}(P).

Then there exist subgroups A ∈ A(T) and L of P such that the following hold:

(a) L is AO_{p}(L)-minimal.

(b) O_{p}(P)A ≤ T ∩L ∈ SyL_{p}(L), and M ∩L is the unique maximal subgroup of L containing AO_{p}(L).

(c) Y ⋬ L, and V_{0} := ⟨Y^{L}⟩ is abelian.

(d) If Y ≤ Z(J(O_{p}(L))), then L and Y satisfies the hypothesis of 4.7 with L in place of P.

Proof: From (ii) [Y, J(T)] = 1 and so J(T) ≤ R and J(R) = J(T). So the assumptions of 4.7 are fulfilled. In particular, V is elementary abelian and there exists A ∈ A(T) with [V, A, A] = 1 ≠ [V, A] and A ≰ O_{p}(P).

Hence we are allowed to apply the L-Lemma 4.3. This gives a subgroup L having the properties (a) – (c) given in 4.3. By 4.3(c) L is not a p-group, and so L is AO_{p}(L)-minimal. This is (a).

According to 4.3(a) there exists g ∈ P such that AO_{p}(L) ≤ T^{g} ∩ L ∈ Syl_{p}(L), and L ∩ M^{g} is the unique maximal subgroup containing AO_{p}(L). Hence replacing A by A^{g^{−1}} and L by L^{g^{−1}} we may assume that (b) holds.

Clearly Y ⋬ L since L ≰ M but N_{P} (Y) ≤ M. Since V is abelian, V_{0} is abelian and (c) holds. From A ≰ O_{p}(P) and (ii) we get that that J(C_{T ∩L}(Y)) ≰ O_{p}(L) and L is C_{T ∩L}(Y)O_{p}(L)-

minimal. Hence 4.7(iii) holds for L and Y. Assume that C_{O_{p}(L)}(Y) ⊴ L. Then also C_{O_{p}(P)}(Y) ⊴ L and thus P = ⟨T, L⟩ ≤ N_{P} (C_{O_{p}(P)}(Y)). This contradicts (i). Hence also 4.7(ii) holds for L and Y, and (d) follows. □

5 F-stability

In this section we explore the following hypothesis:

Hypothesis 5.1 Let p be a prime and H a finite group. Suppose that Y is an elementary abelian p-subgroup of H such that for T ∈ Syl_{p}(N_{H}(Y)) and R := C_{T} (Y) the following hold:

(i) Y ⊴ N_{H}(J(R)).

(ii) Y is F -stable in H.

(iii) Either Y ≤ O_{p}(H) or H is of characteristic p.

This hypothesis is motivated by the following observation:

Lemma 5.2 Let G be a finite group, S ∈ Syl_{p}(G) and J(S) ≤ H ≤ G, and let F(S) be a factorization family for S(S). Suppose that N ∈ F(S) is F -stable.

(a) If Y_{N} ≤ O_{p}(H), then Y := Y_{H} and H satisfy Hypothesis 5.1.

(b) If H is p-constrained and \overline{H} := H/O_{p′}(H), then \overline{Y_{N}} and \overline{H} satisfy Hypothesis 5.1 in place of Y and H.

Proof: Let T ∈ Syl_{p}(H) with J(S) ≤ T. Put Y := Y_{N} and R := C_{T} (Y). Since Y ⊴ S and Y is F -stable, 2.4(a) implies that Y ≤ Ω_{1}Z(J(S)) ≤ H and J(S) = J(T) = J(R). Observe that Y ≤ O_{p}(N_{G}(J(S))) and so by 1.4(b), N_{G}(J(S)) ≤ N. In particular, T ≤ N_{G}(Y) and so T ∈ Syl_{p}(N_{H}(Y)). Now (a) follows.

Assume that H is p-constrained. Then \overline{H} = H/O_{p′}(H) is of characteristic p. By the Frattiniargument, N\overline{H}(\overline{Y})) = \overline{N_{H}(Y)} and N\overline{H}(J(\overline{R})) = \overline{N_{H}(J(R))}. Moreover since Y is F -stable in G, \overline{Y} is F -stable in \overline{H}. Thus Hypothesis 5.1 holds for \overline{Y} and \overline{H}. □

Lemma 5.3 Assume Hypothesis 5.1. Then Y ⊴ T, Y ≤ Z(J(T)), J(R) = J(T), N_{H}(T) ≤ N_{H}(Y) and T ∈ Syl_{p}(H).

Proof: Clearly Y ⊴ T. Thus by 2.4(a), [Y, J(T)] = 1. So J(T) ≤ R and J(T) = J(R). Therefore N_{H}(T) ≤ N_{H}(J(R)) and so by Hypothesis 5.1(i) N_{H}(T) ≤ N_{H}(Y). Hence T ∈ Syl_{p}(H). □

Theorem 5.4 Assume Hypothesis 5.1 and suppose C_{O_{p}(H)}(Y) ⋬ H. Then Y is not 2F -stable in H.

Proof: If any subgroup of H satisfies the conclusion of 5.4 with respect to Y, then also H does. Thus we may assume:

1^{∘} No proper subgroup of H satisfies the hypothesis of 5.4 with respect to Y.

Put

H_{0} = N_{H}(C_{O_{p}(H)}(Y)).

From 5.3 we conclude

2^{∘} N_{H}(T) ≤ N_{H}(Y) ≤ H_{0}, J(R) = J(T), Y ≤ ZJ(R)) and T ∈ Syl_{p}(H_{0}).

Next we show:

3^{∘} Let J(R)O_{p}(H) ≤ H̃ < H. Then C_{O_{p}(H̃)}(Y) ⊴ H̃ and H̃ ≤ H_{0}.

By (2^{∘}) there exists R̃ ∈ Syl_{p}(CH̃(Y)) with J(R) ≤ R̃ and J(R) = J(T) = J(R̃), and so Y ⊴ NH̃(J(R̃)). Since O_{p}(H) ≤ O_{p}(H̃), Y ≤ O_{p}(H̃) or H̃ is of characteristic p. Moreover Y is F -stable in H̃, and so Y and H̃ satisfy Hypothesis 5.1. Now (1^{∘}) shows that C_{O_{p}(H̃)}(Y) ⊴ H̃. Hence C_{O_{p}(H)}(Y) = O_{p}(H) ∩ C_{O_{p}(H̃)}(Y) is normal in H̃ and H̃ ≤ H_{0}.

4^{∘} H_{0} is the unique maximal subgroup of H containing J(R)O_{p}(H), and H is J(R)O_{p}(H)- minimal.

The first statement follows from (3^{∘}). If J(R) ≤ O_{p}(H), then by 5.3, J(R) = J(O_{p}(H)) ⊴ H, and so by Hypothesis 5.1(i), H ≤ N_{H}(Y) ≤ H_{0}, a contradiction. Hence J(R)O_{p}(H) ≰ O_{p}(H), and H is J(R)O_{p}(H)-minimal.

5^{∘} Put W := ⟨Y^{H_{0}}⟩. Then W is elementary abelian.

We will first show that Y ≤ O_{p}(H_{0}). If Y ≤ O_{p}(H), this is obvious. Otherwise H is of characteristic p and by 5.3, O_{p}(H) normalizes Y. So by 2.5 and (2^{∘}) Y ≤ O_{p}(H_{0}) ≤ T. Now (5^{∘}) follows from Hypothesis 5.1(ii) and 2.4(b).

Let W be the set of all p-subgroups D of H satisfying:

(a) W O_{p}(H) ≤ N_{H}(D) ≰ H_{0}.

(b) D = J(D) ≤ H_{0}.

Clearly 1 ∈ W and so W ≠ ∅. Pick D ∈ W such that first |A| is maximal for A ∈ A(D) and then |D| is maximal. Put N := N_{H}(D) and T_{0} := DO_{p}(H) and let T_{1} ∈ Syl_{p}(N ∩ H_{0}). Since T_{0} ≤ O_{p}(N ∩ H_{0}), T_{0} ≤ T_{1}. As W is H_{0}-invariant and by (2^{∘}) T ∈ Syl_{p}(H_{0}), there exists g ∈ H_{0} with W^{g} = W and T_{1}^{g} ≤ T; in particular D^{g} ∈ W. Thus, after replacing D by D^{g} we may assume that T_{1} ≤ T.

6^{∘} Y ≤ Z(J(T_{1})), and if Y ≤ T_{0} then Y ≤ Z(J(T_{0})).

Since T_{1} ≤ T, T_{1} normalizes Y. So (6^{∘}) follows from 2.4(a).

7^{∘} Let U be a p-subgroup of H_{0} containing D. Suppose that W ≤ N_{H}(U) and N_{H}(U) ≰ H_{0}. Then J(U) = D, and if Y ≤ U then Y ≤ Z(D).

Observe that W ≤ N_{H}(U) ≤ N_{H}(UO_{p}(H)). Hence N_{H}(UO_{p}(H)) ≰ H_{0} and J(UO_{p}(H)) ∈ W. Since D ≤ U ≤ UO_{p}(H), the maximal choice of D gives D = J(UO_{p}(H)) = J(U).

Suppose that Y ≤ U, then J(U) = D ≤ T_{1} ≤ N_{H}(Y) and so by 2.4(a), Y ≤ Z(J(U)) = Z(D).

8^{∘} J(T) ≠ D and J(T_{1}) ≠ D.

Suppose J(T) = D. Then by 5.3 and Hypothesis 5.1(i), N ≤ N_{G}(J(T)) ≤ N_{G}(Y) and so N ≤ N_{H}(C_{O_{p}(H)}(Y)) ≤ H_{0}, contrary to the choice of D.

Suppose J(T_{1}) = D. Then N_{T} (T_{1}) ≤ N_{H}(J(T_{1})) = N. So N_{T} (T_{1}) ≤ T_{1}, T = T_{1} and J(T) = J(T_{1}) = D, a contradiction.

9^{∘} Let U be a p-subgroup of H_{0} containing W D. Suppose that J(U) ≠ D or Y ≰ Z(D). Then U is not contained in any H-conjugate of H_{0} other than H_{0}.

Let g ∈ H with U ≤ H_{0}^{g} and U ≤ T_{2} ∈ Syl_{p}(H_{0} ∩ H_{0}^{g}). If J(T_{2}) = D, then also D = J(U) and thus Y ≰ Z(D). Thus either J(T_{2}) ≠ D or Y ≰ Z(D). So (7^{∘}) gives N_{H}(T_{2}) ≤ H_{0}. This implies N_{H_{0}^{g}}(T_{2}) ≤ H_{0} ∩ H_{0}^{g} and so T_{2} ∈ Syl_{p}(H_{0}^{g}). By (4^{∘}), H_{0}^{g} is the unique maximal subgroup of H containing T_{2}. Since T_{2} ≤ H_{0} we get H_{0} = H_{0}^{g}.

10^{∘} T_{1} ∈ Syl_{p}(N), J(T_{1}) ≰ O_{p}(N), and W J(T_{1})D is not contained in any other H-conjugate of H_{0}.

By (8^{∘}) and (7^{∘}) N_{H}(J(T_{1})D) ≤ H_{0}, so N_{N}(T_{1}) ≤ N ∩H_{0} and T_{1} ∈ Syl_{p}(N). If J(T_{1}) ≤ O_{p}(N), then J(T_{1}) = J(O_{p}(N)) and N ≤ N_{H}(J(T_{1})D) ≤ H_{0}, a contradiction.

Put U := W J(T_{1})D. By (8^{∘}), J(U) ≠ D and so the last statement in (10^{∘}) follows from (9^{∘}).

11^{∘} There exists a W J(T_{1})T_{0}-minimal subgroup H_{1} ≤ N such that H_{1} ∩ H_{0} is a maximal subgroup of H_{1} and J(O_{p}(H_{1})) = D.

By definition of W, N ≰ H_{0}. Choose W J(T_{1})T_{0} ≤ H_{1} ≤ N such that H_{1} is minimal with H_{1} ≰ H_{0}. Since H_{1} ≰ H_{0}, N_{H}(O_{p}(H_{1})) ≰ H_{0}. Also W O_{p}(H) normalizes O_{p}(H_{1}) and D ≤ O_{p}(H_{1}). So by (7^{∘}) J(O_{p}(H_{1})) = D. Since J(T_{1}) ≠ D by (8^{∘}) we conclude J(T_{1}) ≰ O_{p}(H_{1}). Hence also W J(T_{1})T_{0} ≰ O_{p}(H_{1}), and H_{1} is W J(T_{1})T_{0}-minimal.

In the following let H_{1} be as in (11^{∘}). Pick W J(T_{1})T_{0} ≤ T_{3} ∈ Syl_{p}(H_{1} ∩ H_{0}). Then H_{1} is T_{3}- minimal and so T_{3} ∈ Syl_{p}(H_{1}). Since T_{3} ≤ N ∩ H_{0} and T_{1} ∈ Syl_{p}(H_{0} ∩ N), there exists g ∈ N ∩ H_{0} with J(T_{1}) ≤ T_{3} ≤ T_{1}^{g}. Hence g normalizes J(T_{1}), D and W, and thus also W J(T_{1})T_{0}. So replacing H_{1} by H_{1}^{g} and T_{3} by T_{3}^{g} we may assume that T_{3} ≤ T_{1} ≤ T.

Case 1 The case Y ≤ O_{p}(H_{1}).

12^{∘} Y and H_{1} satisfy the hypotheses of 4.8.

Since T_{3} ≤ T_{1}, Y ⊴ T_{3}. By (5^{∘}) and (6^{∘}) Y ≤ Z(W J(T_{1})), so H_{1} satisfies Hypothesis 4.1. Hence (8^{∘}) and (11^{∘}) give Hypothesis 4.8(iii), while 2.4(a) gives Hypothesis 4.8(ii).

Assume that C_{O_{p}(H_{1})}(Y) ⊴ H_{1}. As O_{p}(H) ≤ O_{p}(H_{1}), also C_{O_{p}(H)}(Y) ⊴ H_{1}, which contradicts H_{1} ≰ H_{0}. Hence also Hypothesis 4.8(i) holds.

According to (12^{∘}) we are allowed to apply 4.8 to Y and H_{1}. Let L and V be with the properties given there. Since Y ≤ O_{p}(L) ≤ T_{1}, we get from 2.4(a) that Y ≤ Z(J(O_{p}(L)). Thus, by 4.8(d) L and Y satisfy the hypothesis of 4.7.

Since Y is 2F -stable we are in case 4.7(b), so 0 ≠ q(Y, O_{p}(H_{1})) ≤ 2. Thus there exists non-trivial quadratic 2F-offender on Y and the lemma is proved in (Case 1).

Case 2 The case Y ≰ O_{p}(H_{1}).

By our assumption on H, in this case H has characteristic p. Hence also H_{1} has characteristic p since O_{p}(H) ≤ H_{1}. We now apply the L-Lemma 4.3 with W and H_{1} in place of A and P. Then there exists W O_{p}(H_{1}) ≤ L such that

(i) L is W O_{p}(L)-minimal and (ii) there exists g ∈ H_{1} such that L_{0} := H_{0}^{g} ∩ L is the unique maximal subgroup of L containing

W O_{p}(L).

13^{∘} L_{0} = L ∩ H_{0}, Y ≰ Z(D) and J(O_{p}(L)) = D.

Let g as in (ii). Then W D ≤ H_{0} ∩ H_{0}^{g} and Y ≰ Z(D) since D ≤ O_{p}(H_{1}). Hence (9^{∘}) implies H_{0} = H_{0}^{g}; in particular L_{0} = L ∩ H_{0} and L ≰ H_{0}. Now (7^{∘}) also gives J(O_{p}(L)) = D.

According to (13^{∘}) we may assume, after conjugation by a suitable element of H_{0} ∩ H_{1}, that

14^{∘} W O_{p}(L) ≤ T ∩ L ∈ Syl_{p}(L_{0}). In particular O_{p}(L) ≤ T and O_{p}(L) normalizes Y.

By (13^{∘}), Y ≰ ZJ(O_{p}(L)). Since O_{p}(L) normalizes Y, we get from 2.4 that

15^{∘} Y ≰ O_{p}(L).

Put

A := W, B := Y, X := O_{p}(L) ∩ A, E := ⟨X^{L}⟩, Z := O_{p}(L) ∩ B.

By (5^{∘}) A is abelian, and by (14^{∘}) O_{p}(L) normalizes A and B. Moreover, since O_{p}(H_{1}) ≤ O_{p}(L) and H_{1} has characteristic p, [E, O^{p}(L)] ≠ 1. It follows that the hypotheses of 4.5 are satisfied.

By 4.5(a), C_{Y} (E) ≤ Y ∩ O_{p}(L) and so by 4.5(c), [b, Y]C_{Y} (E) = [E, Y]C_{Y} (E) = Y ∩ O_{p}(L) for all b ∈ Y \ O_{p}(L). Moreover, by 4.5(d) [Y, E, E, E] = 1 and by 4.5(e),(f) we have |Y/C_{Y} (E)| ≤ |E/C_{E}(Y)|^{2}. So E is a nearly quadratic 2F -offender on Y. Hence the lemma also holds in (Case 2).

□

Lemma 5.5 Assume Hypothesis 5.1. Suppose that Y ⋬ H and Y is 2F -stable. Then Ω_{1}Z(T) ⋬ H.

Proof: Let T ≤ P ≤ H and P be minimal with Y ⋬ P. By 5.3 N_{H}(T) ≤ N_{H}(Y), so T ⋬ P and P is T -minimal. Put

Q := C_{O_{p}(P)}(Y), V_{0} := Ω_{1}(Z(Q)), V := C_{V_{0}}(O_{p}(P)), \overline{P} := P/C_{P} (V).

If Z(T) ≰ O_{p}(P), Z(T) ⋬ P. So we may assume Ω_{1}Z(T) ≤ O_{p}(P) and thus Ω_{1}Z(T) = C_{V} (T). By 5.4 Q ⊴ P. Since either Y ≤ O_{p}(H) ≤ O_{P} (P) or P is of characteristic p, 2.5 implies Y ≤ O_{p}(P). Thus Y ≤ V_{0}. Since Y ⋬ P, we get that [V_{0}, O^{p}(P)] ≠ 1. By 5.3, J(R) = J(T) and so by Hypothesis 5.1(i), J(R) ≰ O_{p}(P). Hence 2.3 shows that [O^{p}(P), J(R)] = O^{p}(P). Since J(R) centralizes Y, [O_{p}(P), J(R)] ≤ O_{p}(P) ∩ J(T) ≤ Q and so [O_{p}(P), O^{p}(P)] ≤ Q. The P × Q-Lemma yields [V, O^{p}(P)] ≠ 1.

Again 2.3 gives C_{T} (V) = O_{p}(P) and O_{p}(\overline{P}) = 1. Moreover \overline{J(T)} ≠ 1 since J(R) ≰ O_{p}(P). Hence \overline{P} and V satisfy the hypothesis of [BHS, 5.6]. It follows that [C_{V} (T), P] ≠ 1. Since C_{V} (T) = Ω_{1}Z(T) and P = ⟨T^{P} ⟩ we conclude that Ω_{1}Z(T) ⋬ P and so also Z(T) ⋬ H. □

6 The Proof of Theorems 1.5 – 1.8

Recall that Theorems 1.3 and 1.4 have been proved in Section 3.

Proof of Theorem 1.5:

(a): Observe that N = N_{G}(Y_{N}) by 1.3. Suppose Y_{N} ≤ O_{p}(M). If M = N_{G}(C) for 1 ≠ C char B(S), then 1.5(a) follows from 1.4(b). If Ω_{1}Z(S) ⊴ M, then 5.2(a) shows that Y_{N} and M satisfy Hypothesis 5.1. Hence 5.5 gives Y_{N} ⊴ M and so M ≤ N.

(b): Put \overline{M} := M/O_{p′}(M). Then 5.2(b) shows that \overline{Y_{N}} and \overline{M} satisfy Hypothesis 5.1. Thus 5.5 gives \overline{Y_{N}} ⊴ \overline{M}. By the Frattini-argument M = O_{p′}(M)N_{M}(Y_{N}) = O_{p′}(M)(M ∩ N).

(c): Let B(S) ≤ H ≤ G and H be p-constrained with H ≠ O_{p′}(H)(H ∩ N), and let B(S) ≤ T ∈ Syl_{p}(N_{H}(Y_{N})). Put \overline{H} := H/O_{p′}(H). Then again 5.2(b) shows that \overline{Y_{N}} and \overline{H} satisfy Hypothesis 5.1. Hence by 5.4, C_{O_{p}(\overline{H})}(\overline{Y_{N}}) ⊴ H and by 2.5, \overline{Y_{N}} ≤ O_{p}(\overline{H}). From 5.5 applied to N\overline{H}(Ω_{1}Z(\overline{T})) we get \overline{Y_{N}} ⊴ N\overline{H}(Ω_{1}Z(\overline{T})). Recall that Y_{N} ≤ Ω_{1}Z(J(S)) ≤ O_{p}(N_{G}(J(S)). Thus by 1.4(b), N_{G}(J(S)) ≤ N. Since B(T) = B(S) we have J(T) = J(S) and so \overline{Y_{N}} ⊴ C^{∗∗}(\overline{H}, \overline{T}). By the Frattini Argument N\overline{H}(\overline{Y_{N}}) = \overline{N_{H}(Y_{N})} = \overline{H ∩ N}. Hence also C^{∗∗}(\overline{H}, \overline{T}) ≤ \overline{H ∩ N}.

Proof of Theorem 1.6:

Let P ∈ F(S). By 1.3 P is a p-local subgroup of G. Let L be a maximal p-local subgroup containing P. By 2.2(c) Y_{P} ≤ Y_{L} and so P ≪ L. Hence by 3.4 P = L.

Suppose that N ∈ F(S) is 2F -stable. Let M = N_{G}(C) for 1 ≠ C char B(S) or M = N_{G}(Ω_{1}Z(S)). Then S ≤ M, so M has characteristic p since G is of parabolic characteristic p. Hence 1.5(a) implies M ≤ N.

Let H ∈ S(B(S)) and B(S) ≤ T ∈ Syl_{p}(H). Then B(S) = B(T) and so N_{H}(C) ≤ N for 1 ≠ C char B(T). Also T ≤ S^{g} for some g ∈ N_{G}(B(S)) ≤ N and Ω_{1}Z(S^{g}) ≤ J(S) ≤ T, so Ω_{1}Z(S^{g}) ≤ Z(T) and C_{H}(Ω_{1}Z(T)) ≤ C_{G}(Ω_{1}Z(S^{g})) ≤ N^{g} = N. Thus C^{∗}(H, T) ≤ H ∩ N.

Proof of Corollary 1.7:

By 1.6 the members of F(S) are maximal p-local subgroups. We may assume that there exists a 2F -stable N ∈ F(S).

Let L be a maximal p-local subgroup containing S with L ≰ N and choose M ∈ F(S) with L ≪ M. Then L ≤ C_{G}(Y_{L})M. On the other hand, by 2.2(c) Ω_{1}Z(S) ≤ Y_{L} and so by 1.6 C_{G}(Y_{M}) ≤ C_{G}(Ω_{1}Z(S)) ≤ N. Since L ≰ N we conclude that M ≰ N and M ≠ N. By 1.4 N is the only member of F(S) which is F -stable. Hence M is not F -stable.

Proof of Theorem 1.8:

Let P be the semi-direct product of G and V. Then O_{p}(P) = V and [V, O^{p}(P)] ≠ 1. Let A be an offender on V such that |A||C_{V} (A)| is maximal. Because of [KS, 9.2.3] we may assume that A is quadratic on V Hence 4.6 implies |A/C_{A}(V)| = |V |, and 1.8 follows.

References

[BHS] D. Bundy, N. Hebbinghaus, B. Stellmacher, The Local C(G, T)-Theorem, to appear J. Algebra.

[GLM] R. M. Guralnick, R. Lawther, G. Malle, The 2F -Modules of Nearly Simple Groups, preprint.

[GM1] R. M. Guralnick, G. Malle, Classification of 2F -Modules, I, J. Algebra 257 (2002), 348– 372.

[GM2] R. M. Guralnick, G. Malle, Classification of 2F -Modules, II, Finite Groups 2003, 117–183.

[KS] H. Kurzweil, B. Stellmacher, The Theory of Finite Groups, Springer Universitext, New York, 2004, xii+387 pp.

[L] R. Lawther, 2F -Modules, Abelian Sets of Roots and 2-ranks, preprint.

[MSS] U. Meierfrankenfeld, B. Stellmacher, G. Stroth, The Structure Theorem, in preparation.

[PPS] C. Parker, G. Parmeggiani, B. Stellmacher, The P!-Theorem, J. Algebra 263 (2003), no. 1, 17–58.

[Ste] B. Stellmacher, On the 2-local Structure of Finite Groups, in Groups, Combinatorics and Geometry, LMS Lecture Notes Series 165 (1992), Cambridge University Press.