The Local C(G, T) Theorem

D. Bundy, N. Hebbinghaus, B. Stellmacher

August 23, 2005

1 Introduction

In this article all groups considered are assumed to be finite. Moreover G always denotes a group and p always a prime.

We define A(G) to be the set of elementary abelian p-subgroups of G of maximal order and Ω(G) to be the subgroup generated by the elements of order p of G. Then

J(G) := ⟨A | A ∈ A(G)⟩

is the Thompson subgroup of G (with respect to p), and

B(G) := ⟨C_{T} (Ω(Z(J(T)))) | T ∈ Syl_{p}(G)⟩

is the Baumann subgroup of G (with respect to p).

Definition 1.1 Let p divide the order of G, T ∈ Syl_{p}(G) and S ≤ T. Then

\begin{aligned}C(G, S) & := ⟨N_{G}(C) | 1 ≠ C char S⟩, \\ C^{∗}(G, T) & := ⟨C_{G}(Ω(Z(T))), C(G, B(T))⟩, \\ C^{∗∗}(G, T) & := ⟨C_{G}(Ω(Z(T))), N_{G}(J(T))⟩. \end{aligned}

Notice that every characteristic subgroup of B(T) is characteristic in T and J(T) is characteristic in B(T). In particular

C^{∗∗}(G, T) ≤ C^{∗}(G, T) ≤ C(G, T).

Definition 1.2 A group G is of characteristic p if

C_{G}(O_{p}(G)) ≤ O_{p}(G) (or equivalently F^{∗}(G) = O_{p}(G)).

In this paper we will classify those groups G of characteristic p that are not equal to C(G, T) with respect to some Sylow p-subgroup T; a result called the Local C(G, T)- Theorem.

The investigation of groups of characteristic p in which G ≠ C(G, T) is a natural extension of work on failure of Thompson factorization as first studied by Glauberman [8] in response to the factorization theorems of Thompson [17]. Indeed Glauberman’s Theorem is similar to that of our C^{∗∗}(G, T)-Theorem for minimal parabolic subgroups (see Theorem 1.5) in the case when G is p-solvable but without the assumption that G is minimal parabolic.

The Local C(G, T)-Theorem in the case p = 2 was proven by Aschbacher [1] and there are some key features of Aschbacher’s proof which we have reformulated for use in our proof. In particular, B(T)-blocks are a generalization of his short groups to the case of p any prime, together with the extra condition that they are normalized by B(T). Aschbacher uses the word block for a short subnormal subgroup.

Some of the properties of B(T)-blocks resemble those of components and these are proven in Chapter 6. For example, distinct subnormal B(T)-blocks commute (6.11). Furthermore, our notations O_{G}(V) and A_{G}(V) are essentially the same as P(G, V) and P^{∗}(G, V) of Aschbacher.

An alternative proof for p = 2 was also given by Gorenstein and Lyons [9]. Their proof avoids the use of some deep results needed in Aschbacher’s proof. Instead it requires the K-group hypothesis (that any simple section of G is one of the known finite simple groups), which is sufficient for the purposes of the classification of the finite simple groups.

Our proof works for all primes p and does neither use the K-group assumption nor the deep results used in Aschbacher’s proof. In fact, it is more or less self contained.

Our result can be considered as part of a project of Meierfrankenfeld et. al. [13] and we will use standard concepts from this project. In particular, the name characteristic p for groups with the property C_{G}(O_{p}(G)) ≤ O_{p}(G) and the L-Lemma originate there. Our abstract definition of a minimal parabolic group is also used extensively in this project, but was originally an idea of McBride.

The Baumann subgroup and the Baumann Argument (3.7) first appeared in (2.11.1) of [2], but we prefer to quote [15], where the result is explicitly stated in the form we require. This result is used to show that certain subgroups satisfy the hypothesis of a pushing up result [16] which was originally proven by Glauberman and Niles [14] and independently for the case p = 2 by Baumann [3].

Groups generated by conjugacy classes of transvections were classified by McLaughlin [10], [11] and some of our results in Section 4 follow easily from this classification, but we prefer to give an independent proof tailored to our particular situation.

The results of Section 2 are elementary and mainly well-known. We have given explicit proofs rather than searching for original references in order to keep things reasonably self contained.

To state the main result we need two further definitions.

Definition 1.3 The symmetric group of degree m is denoted by S_{m}. Let X be a group and W be a finite simple GF (p)X-module. If X ≅ S_{m}, m ≥ 3, then W is a natural S_{m}-module (for X), if p = 2 and W is isomorphic to the unique non-trivial simple section of the GF (2)S_{m}-permutation module.

If X ≅ SL_{2}(p^{m}), then W is a natural SL_{2}(p^{m})-module (for X), if W is irreducible, F := End_{X}(W) ≅ GF (p^{m}), and W is a 2-dimensional F X-module.

Moreover, for A_{m} and SL_{2}(p^{m})′ rather than S_{m} and SL_{2}(p^{m}) the corresponding module is called a natural A_{m}-module and a natural SL_{2}(p^{m})′-module, respectively.

It is easy to see that every finite simple GF (2)S_{m}-module with |W/C_{W} (t)| = 2 for a transposition t ∈ S_{m} is a natural S_{m}-module.

Definition 1.4 Let T ∈ Syl_{p}(G). A subgroup E ≤ G is a B(T)-block of G if for W := Ω(Z(O_{p}(E))):

(i) E = O^{p}(E) = [E, B(T)], [O_{p}(E), E] = O_{p}(E), and [E, Ω(Z(T))] ≠ 1.

(ii) E/O_{p}(E) ≅ SL_{2}(p^{n})′ or p = 2 and E/O_{2}(E) ≅ A_{2m+1}, and W/C_{W} (E) is a natural SL_{2}(p^{n})′- resp. A_{2m+1}-module for E/O_{p}(E).

(iii) O_{p}(E) = W, or

(1) p = 3, and O_{3}(E)/W is a natural SL_{2}(3^{n})′-module for E/O_{3}(E),

(2) O_{3}(E)′ = Φ(O_{3}(E)) = Z(E) = C_{W} (E) and |Z(E)| = 3^{n}, and

(3) no element of B(T) \ C_{B(T)}(W) acts quadratically on O_{3}(E)/Z(E).

If E/O_{p}(E) ≅ SL_{2}(p^{n})′, then E is a linear block, and in the other case E is a symmetric block. Moreover, if (1) – (3) in (iii) hold, then E is an exceptional block.

We will prove the following theorem.

Theorem 1.5 (Local C^{∗}(G, T)-Theorem) Let G be of characteristic p with T ∈ Syl_{p}(G) such that G ≠ C^{∗}(G, T). Then there exist B(T)-blocks G_{1}, …, G_{r} of G such that the following hold:

(a) {G_{1}, …, G_{r}}^{G} = {G_{1}, …, G_{r}}.

(b) [G_{i}, G_{j}] = 1 for i ≠ j.

(c) G = C^{∗}(G, T)G_{0}, where G_{0} := ∏_{i=1}^{r} G_{i}.

(d) Every B(T)-block of G that is not in C^{∗}(G, T) is contained in one of the B(T)- blocks G_{1}, …, G_{r}.

(e) C^{∗}(G, T) ∩ G_{0} = ∏_{i=1}^{r}(C^{∗}(G, T) ∩ G_{i}). Moreover either

(i) G_{i}/O_{p}(G_{i}) ≅ SL_{2}(p^{m}), p^{m} > 3, and C^{∗}(G, T) ∩ G_{i} = N_{G_{i}}(T ∩ G_{i}), or

(ii) p = 2, G_{i}/O_{2}(G_{i}) ≅ A_{2m+1}, and (C^{∗}(G, T) ∩ G_{i})/O_{2}(G_{i}) ≅ A_{2m},

(iii) p = 3, G_{i}/O_{3}(G_{i}) ≅ SL_{2}(3)′ and (C^{∗}(G, T) ∩ G_{i})/O_{3}(G_{i}) = Z(G_{i}/O_{3}(G_{i})).

Corollary 1.6 (Local C(G, T)-Theorem) Let G be of characteristic p with T ∈ Syl_{p}(G) such that G ≠ C(G, T). Then G has the same structure as given in Theorem 1.5 with the additional restriction that if G_{i} is a symmetric block, then G_{i}/O_{2}(G_{i}) ≅ A_{2^{n}+1}.

It is easy to see that under the assumption of Theorem 1.5 every proper subgroup L with B(T) ≤ L and L ≰ C^{∗}(G, T) satisfies the hypothesis of 1.5 (see 2.3). Hence, those groups G, where C^{∗}(G, T) is the unique maximal subgroup containing B(T), are the basis for an induction on the order of G. This leads to a class of groups that plays the same role for groups of local characteristic p as the class of minimal parabolic groups for groups of Lie type in characteristic p (see [13]).

Definition 1.7 Let T ∈ Syl_{p}(G). Then G is a minimal parabolic group (with respect to p), if T is not normal in G and there is a unique maximal subgroup of G containing T.

The restricted structure of minimal parabolic groups allows us to prove a Local C^{∗∗}(G, T)-Theorem that is of interest on its own:

Theorem 1.8 (Local C^{∗∗}(G, T)-Theorem for Minimal Parabolic Groups) Let G be a minimal parabolic group of characteristic p with T ∈ Syl_{p}(G) such that G ≠ C^{∗∗}(G, T), and let V := Ω(Z(O_{p}(G))) and \overline{G} := G/C_{G}(V). Then there exist subgroups E_{1}, …, E_{r} of G such that

(a) \overline{G} = \overline{J(G)T} and \overline{J(G)} = \overline{E}_{1} × ⋯ × \overline{E}_{r},

(b) \overline{T} acts transitively on \{\overline{E}_{1}, …, \overline{E}_{r}\},

(c) V = C_{V} (\overline{E}_{1} × … × \overline{E}_{r}) ∏_{i=1}^{r}[V, E_{i}], with [V, E_{i}, E_{j}] = 1,

(d) \overline{E}_{i} ≅ SL_{2}(p^{n}) or p = 2 and \overline{E}_{i} ≅ S_{2^{n}+1}, for some n ∈ ℕ, and

(e) [V, E_{i}]/C_{[V,E_{i}]}(E_{i}) is a natural module for E_{i}.

As a corollary of the Local C^{∗}(G, T)-Theorem we get a pushing up result for minimal parabolic groups.

Corollary 1.9 (Pushing Up Theorem for Minimal Parabolic Groups) Let G be a minimal parabolic group of characteristic p with T ∈ Syl_{p}(G). Suppose that neither any non-trivial characteristic subgroup of B(T) nor Ω(Z(T)) is normal in G. Then G satisfies the conclusion of the Local C(G, T)-Theorem. Moreover C^{∗}(G, T) = (C^{∗}(G, T) ∩ ∏_{i=1}^{r} G_{i})T.

2 Preliminary Results

Lemma 2.1 Let D be a conjugacy class of subgroups of G and A and B be subgroups of G. Suppose that G = ⟨D⟩ and

D = {X ∈ D | X ≤ A} ∪ {X ∈ D | X ≤ B}.

Then A = G or B = G.

Proof. Let

D_{0} := {X ∈ D | X ≰ A} and D := ⟨D_{0}⟩.

We may assume that A ≠ G, so D_{0} ≠ ∅. Clearly D ≤ B and ⟨A, D⟩ ≤ N_{G}(D). Moreover, every element of D is a subgroup of A or D, whence G = ⟨D⟩ ≤ N_{G}(D). Since D is a conjugacy class of G and D_{0} ≠ ∅, this gives G = D = B. □

Lemma 2.2 Let G be of characteristic p and L ≤ G. Any of the following conditions implies that L is of characteristic p :

(a) L ⊴⊴ G.

(b) O_{p}(G) ≤ L.

(c) L ⊴⊴ ⟨L, O_{p}(G)⟩.

(d) O_{p}(G) normalizes L.

Proof. (a): Since L ⊴⊴ G, F^{∗}(L) ≤ F^{∗}(G) = O_{p}(G).

(b): O_{p}(G) ≤ O_{p}(L), so C_{L}(O_{p}(L)) ≤ C_{G}(O_{p}(G)) ≤ O_{p}(G) ≤ O_{p}(L).

(c): By (b) ⟨L, O_{p}(G)⟩ has characteristic p. Thus (a) (with ⟨L, O_{p}(G)⟩ in place of G) shows that L has characteristic p.

(d): L ⊴ LO_{p}(G). So (d) follows from (c).

□

Lemma 2.3 Let G be of characteristic p, T ∈ Syl_{p}(G), and Q ⊴ T with C_{T} (Q) ≤ Q. Suppose that L and P are subgroups of G such that Q ≤ L and B(T) ≤ T_{0} ∈ Syl_{p}(P). Then the following hold:

(a) C_{G}(Q) ≤ Q.

(b) L is of characteristic p.

(c) P is of characteristic p.

(d) C^{∗}(P, T_{0}) ≤ C^{∗}(G, T).

(e) If P is minimal with respect to T_{0} ≤ P and P ≰ C^{∗}(G, T), then P is a minimal parabolic of characteristic p with C^{∗}(P, T_{0}) ≠ P.

Proof. (a): Let D := C_{G}(Q). Since C_{T} (Q) ≤ Q and O_{p}(G) ≤ T, C_{O_{p}(G)}(Q) ≤ Q ≤ C_{G}(D). So by the P × Q-Lemma, O^{p}(D) centralizes O_{p}(G). Hence D is a p-group since G is of characteristic p. As T normalizes D, D ≤ T and so D ≤ C_{T} (Q) ≤ Q.

(b): Let L_{0} := ⟨Q^{L}⟩. Since Q ⊴ T, O_{p}(G) normalizes Q and so also L_{0}. Hence by 2.2(d) L_{0} is of characteristic p. Let C := C_{L}(O_{p}(L)). Then C ≤ C_{L}(O_{p}(L_{0})) and thus

[Q, C] ≤ L_{0} ∩ C ≤ C_{L_{0}}(O_{p}(L_{0})) ≤ O_{p}(L_{0}) ≤ O_{p}(L) and [Q, C, C] = 1.

Hence C normalizes QO_{p}(L), so [QO_{p}(L), C, C] = 1 and [Q, O^{p}(C)] = 1. By (a) O^{p}(C) = 1, and C is a p-group. Thus C ≤ O_{p}(L), and L is of characteristic p.

(c): Observe that B(T) ⊴ T and C_{T} (B(T)) ≤ B(T). Hence (c) follows from (b). (d): Let T_{0} ≤ T̃ ∈ Syl_{p}(G). Then T̃ ≤ N_{G}(B(T)) ≤ C^{∗}(G, T), so C^{∗}(G, T) =

C^{∗}(G, T̃). Thus we may assume that T_{0} ≤ T. Then

Ω(Z(T)) ≤ C_{T} (B(T)) ≤ B(T) ≤ T_{0},

and so Ω(Z(T)) ≤ Ω(Z(T_{0})). It follows that

C_{P} (Ω(Z(T_{0}))) ≤ C_{P} (Ω(Z(T))) ≤ C_{G}(Ω(Z(T))) ≤ C^{∗}(G, T).

Since B(T) = B(T_{0}), we conclude that C^{∗}(P, T_{0}) ≤ C^{∗}(G, T).

(e): From (c) and (d) we get that P is of characteristic p and

C^{∗}(P, T_{0}) ≤ P ∩ C^{∗}(G, T) ≠ P.

The minimal choice of P shows that P ∩ C^{∗}(G, T) is the unique maximal subgroup of P containing T_{0}. As N_{P} (T_{0}) ≤ N_{P} (B(T)) ≤ P ∩ C^{∗}(G, T), P is a minimal parabolic subgroup of G. □

Lemma 2.4 Let G = QN, where N is a normal subgroup of G and Q is a non-abelian 2-subgroup with Q ∩ N = 1. Suppose that there exists 1 ≠ t ∈ Z(Q) ∩ Q′ such that

(∗)  C_{N}(Q) = C_{N}(t).

Then [N, Q] is solvable of odd order.

Proof. There exists S ∈ Syl_{2}(N) such that Q ≤ N_{G}(S). Let g ∈ N such that a := t^{g} ∈ tS and [t, a] = 1. Then

ta ∈ C_{S}(t) =^{(∗)} C_{S}(Q),

so [Q, a] = 1, since t ∈ Z(Q). Now (∗) implies

Q ≤ C_{G}(a) = Q^{g} × C_{N}(Q^{g}).

Let Q_{0} be the projection of Q in C_{N}(Q^{g}). Then t centralizes Q_{0}, so by (∗) also [Q, Q_{0}] = 1. It follows that Q′ ≤ Q^{g} ∩ Q and [Q′, g] ≤ Q′Q^{′g} ∩ N ≤ Q^{g} ∩ N = 1. But now t = a since t ∈ Q′.

We have shown that t itself is the only conjugate of t in ⟨t⟩S that commutes with t. It follows that t is not conjugate in G to any other element of ⟨t⟩S. Hence, Glauberman’s Z^{∗}-Theorem [6] together with (∗) implies that [N, Q] is a group of odd order, and the Theorem of Feit-Thompson [5] yields the desired result. □

Lemma 2.5 Let G be of characteristic p. Suppose that there exist subgroups E ≤ G and N ⊴ G with [O^{p}(N), E] = 1, [O_{p}(G), E] ≤ E, and E = O^{p}(E). Then E ⊴ N_{G}(EN).

Proof. Let E_{0} := E[E, N_{G}(EN)] = ⟨E^{N_{G}(EN)}⟩ and R := E_{0} ∩ N. Then

(∗)  E_{0} = ER and O^{p}(R) ≤ Z(E_{0}).

Note that O_{p}(G) normalizes E_{0}. Hence by 2.2 E_{0} has characteristic p, so by (∗) O^{p}(R) ≤ O_{p}(E_{0}) and O^{p}(R) = 1. Thus R is a p-group. It follows that R ≤ O_{p}(N) ≤ O_{p}(G). Then O^{p}(E_{0}) = E_{0} and [O_{p}(G), E] ≤ E imply R ≤ E and E ⊴ N_{G}(EN). □

Lemma 2.6 Let E be a group, Q := O_{3}(E), W := Ω(Z(Q)) and Z := C_{W} (E). Suppose that the following hold:

(i) E/Q ≅ SL_{2}(3^{n}).

(ii) Q/W and W/Z are natural SL_{2}(3^{n})-modules for E/Q.

(iii) Z = Φ(Q) = Q′ = Z(E) and |Z| = 3^{n}.

Then the image of C_{Aut(Q)}(Z) in Aut(Q/W) is isomorphic to SL_{2}(3^{n}).

Proof. Let W_{0} := [W, E] and q := 3^{n}. Then W = Z×W_{0}, and \overline{Q} := Q/W_{0} is a special group of order q^{3}. Let W ≤ A ≤ Q and T ∈ Syl_{3}(E) such that A/W = Z(T/W). Pick a ∈ A \ W. By (ii) and (iii) [a, Q] = ⟨[a, Q]^{E}⟩ = Q′ = Z and thus |Q/C_{Q}(a)| = q. As also \overline{C_{Q}(a)} is normalized by T and Q/W is a natural SL_{2}(q)-module, we get that \overline{A} = \overline{C_{Q}(a)} and thus A = C_{Q}(a); in particular A is abelian.

Let D := {A^{e} | e ∈ E}. For B ∈ D we have:

(∗)  C_{Q}(b) = B for b ∈ B \ W.

Moreover |D| = q + 1, and the images of the elements of D form a partition of Q/W. This latter property together with (∗) shows that the elements in D are the only abelian subgroups of order q^{4} in Q.

Pick A, B ∈ D, A ≠ B. Then (∗) implies

[a, b] ≠ 1 for all a ∈ A \ W and b ∈ B \ W.

The action of E on \overline{Q} shows that C_{E}(\overline{A}/\overline{Z}) acts regularly on D \ \{A\}.

Now let α ∈ Y := C_{Aut(Q)}(Z). Assume that α centralizes \overline{A}/\overline{Z}. If α normalizes B, then for b ∈ B and a ∈ A

[b, a] = [b, a]α = [bα, a],

so b^{−1}(bα) ∈ W. Hence α centralizes AB/W = Q/W, and so

C_{Y} (\overline{A}/\overline{Z}) ∩ N_{Y} (\overline{B}) = C_{Y} (Q/W).

With a similar argument C_{Y} (\overline{a}\overline{Z}/\overline{Z}) ∩ N_{Y} (\overline{B}) ≤ C_{Y} (\overline{B}/\overline{Z}), so

C_{Y} (\overline{a}\overline{Z}/\overline{Z}) ∩ N_{Y} (\overline{B}) = C_{Y} (\overline{B}/\overline{Z}) ∩ N_{Y} (\overline{A}) = C_{Y} (Q/W).

It follows that |Y/C_{Y} (Q/W)| ≤ q(q − 1)(q + 1), because there are (q − 1)(q + 1) choices for \overline{a}\overline{Z} and then q choices for B with \overline{a} ∉ \overline{B}. As E induces SL_{2}(q) on Q/W, we are done.

□

Definition 2.7 Let V be a finite dimensional GF (p)G-module. Then O_{G}(V) is the set of subgroups A of G such that:

(i) [V, A] ≠ 1,

(ii) |A/C_{A}(V)||C_{V} (A)| ≥ |A^{∗}/C_{A^{∗}}(V)||C_{V} (A^{∗})| for all subgroups A^{∗} of A, and

(iii) A/C_{A}(V) is an elementary abelian p-group.

Moreover

O_{G}^{∗}(V) := {A ∈ O_{G}(V) | |A/C_{A}(V)||C_{V} (A)| > |V |}.

Suppose that O_{G}(V) ≠ ∅. Then

m_{G}(V) := max{|A/C_{A}(V)||C_{V} (A)| | A ∈ O_{G}(V)},

and A_{G}(V) is the set of minimal (by inclusion) elements of the set

{A ∈ O_{G}(V) | |A/C_{A}(V)||C_{V} (A)| = m_{G}(V)}.

Observe that property (ii) above with A^{∗} = 1 gives m_{G}(V) ≥ |V |.

Lemma 2.8 Let V be a finite dimensional GF (p)G-module, V_{0} ≤ C_{V} (O^{p}(G)) be a GF (p)G-submodule, and W ≤ V. Then the following hold for A ∈ O_{G}(V):

(a) |W/C_{W} (A)| ≤ |A/C_{A}(W)|.

(b) Let A ∈ O_{G}^{∗}(V). Then |W/C_{W} (A)| < |A/C_{A}(W)| or C_{A}(W) ∈ O_{G}^{∗}(V).

(c) A ∈ O_{N_{G}(C_{V} (B))}(C_{V} (B)) for all B ≤ A with [C_{V} (B), A] ≠ 1.

(d) Let O_{p}(G/C_{G}(V)) = 1. Then O_{G}(V/V_{0}) ≠ ∅ if O_{G}(V) ≠ ∅, and O_{G}^{∗}(V/V_{0}) ≠ ∅ if O_{G}^{∗}(V) ≠ ∅.

(e) Let V be an elementary abelian normal subgroup of G. Then {A ∈ A(G) | [A, V] ≠ 1} ⊆ O_{G}(V).

Proof. (a), (b) and (c): Set A_{0} := C_{A}(W). By the definition of O_{G}(V)

|A||C_{V} (A)| ≥ |A_{0}||C_{V} (A_{0})| ≥ |A_{0}||W C_{V} (A)| = |A_{0}||W ||C_{V} (A)||C_{W} (A)|^{−1},

and thus |A/A_{0}| ≥ |W/C_{W} (A)|.

Moreover, if A ∈ O_{G}^{∗}(V) and |A/A_{0}| = |W/C_{W} (A)|, then |A/C_{A}(V)||C_{V} (A)| = |A_{0}/C_{A}(V)||C_{V} (A_{0})| = m_{G}(V) > |V | and A_{0} ∈ O_{G}^{∗}(V).

Assume now that W = C_{V} (B) for some B ≤ A and set B^{∗} := C_{A}(W), so B ≤ B^{∗} and C_{V} (B^{∗}) = C_{W} (B^{∗}). Then C_{W} (A^{∗}) = C_{W} (A^{∗}B^{∗}) = C_{V} (A^{∗}B^{∗}) for every A^{∗} ≤ A, so

|A||C_{W} (A)| = |A||C_{V} (A)| ≥ |A^{∗}B^{∗}||C_{V} (A^{∗}B^{∗})| ≥ |A^{∗}B^{∗}||C_{W} (A^{∗})|

and

|A/B^{∗}||C_{W} (A)| ≥ |A^{∗}B^{∗}/B^{∗}||C_{W} (A^{∗})| = |A^{∗}/A^{∗} ∩ B^{∗}||C_{W} (A^{∗})|.

Hence (c) follows.

(d): Let \overline{V} := V/V_{0}. Observe that C_{A}(V) = C_{A}(\overline{V}), since O_{p}(G/C_{G}(V)) = 1 and that |\overline{V} /C\overline{V} (A)| ≤ |V/C_{V} (A)|.

(e): Let A ∈ A(G). Then the maximality of |A| gives for every A^{∗} ≤ A,

\begin{aligned}|A| = |AC_{V} (A)| & = |A||C_{V} (A)||V ∩ A|^{−1} ≥ |A^{∗}C_{V} (A^{∗})| \\ & = |A^{∗}||C_{V} (A^{∗})||V ∩ A^{∗}|^{−1} ≥ |A^{∗}||C_{V} (A^{∗})||V ∩ A|^{−1}, \end{aligned}

and thus with A^{∗}C_{A}(V) in place of A^{∗}

|A/C_{A}(V)||C_{V} (A)| ≥ |A^{∗}C_{A}(V)/C_{A}(V)||C_{V} (A^{∗})| = |A^{∗}/C_{A^{∗}}(V)||C_{V} (A^{∗})|.

Hence A ∈ O_{G}(V) if [V, A] ≠ 1. □

Notation 2.9 In the following six lemmas we will give some elementary facts about S_{n} in its action on a natural GF (2)-module. For this purpose we fix the following notation.

Let G = S_{n}, n > 1, and V^{∗} be a GF (2)G-permutation module (written multiplicatively); so there exists a basis Ω := {v_{1}, …, v_{n}} that is permuted by G. We set

W := ⟨v_{i}v_{j} | 1 ≤ i, j ≤ n⟩ and V_{0} := ⟨∏_{i=1}^{n} v_{i}⟩.

If n is odd, then V := W is a natural GF (2)G-module, and if n is even, then V := W/V_{0} is a natural GF (2)G-module. Furthermore we fix T ∈ Syl_{2}(G), and Y is the subgroup generated by the transpositions contained in T.

Lemma 2.10 Let G = S_{n}, n ≥ 4. Then either

(a) n is even, and N_{G}(Y) is transitive on the transpositions of G that are not in Y, or

(b) n is odd, and N_{G}(Y) has two orbits on the transpositions not in Y. The elements of one orbit have a fixed point in common with Y and the elements of the other orbit do not.

Proof. This is an elementary calculation in S_{n}. □

Lemma 2.11 Let G = S_{n}, n ≥ 5, and V be a natural GF (2)G-module. Then ⟨N_{G}(Y), C_{G}(C_{V} (T))⟩ ≅ S_{n−1} if n is odd, and G = ⟨N_{G}(Y), C_{G}(C_{V} (T))⟩ if n is even.

Proof. Set M := ⟨N_{G}(Y), C_{G}(C_{V} (T))⟩. Suppose first that n is odd. Then V^{∗} = V_{0}×V and

C_{V^{∗}}(T) = C_{V} (T)×V_{0},

so C_{G}(C_{V} (T)) = C_{G}(C_{V^{∗}}(T)).

There exists a unique v ∈ Ω such that v ∈ C_{V^{∗}}(Y). This element is centralized by N_{G}(Y), and thus also by T. It follows that ⟨C_{G}(C_{V^{∗}}(T)), N_{G}(Y)⟩ fixes v; in particular M ≠ G. Since there are transpositions in C_{G}(C_{V^{∗}}(T)) that are not in Y, 2.10 shows that M contains all transpositions that fix v. Hence M ≅ S_{n−1}.

Suppose that n is even. It suffices to show that M contains a transposition that is not in Y. Since then by 2.10 M contains all the transpositions of G, so M = G.

Let Ω_{1}, …, Ω_{r} be the T -orbits of Ω, and let Λ_{1}, …, Λ_{k} be the proper subsets of Ω with [∏_{v∈Λ_{i}} v, T] ≤ V_{0}. Set

o_{i} := ∏_{v∈Ω_{i}} v, i = 1, …, r, and ℓ_{i} := ∏_{v∈Λ_{i}} v, i = 1 …, k.

Then

C_{W} (T) = ⟨o_{1}, …, o_{r}⟩ and C_{V} (T) = ⟨ℓ_{1}, …, ℓ_{k}⟩/V_{0}.

Assume first that C_{G}(C_{V} (T)) = C_{G}(C_{W} (T)). Since n ≥ 5, we may assume that |Ω_{1}| ≥ 4. Hence there exists a transposition d ∈ N_{G}(Ω_{1}) \ C_{G}(Ω_{1}) with d ∉ Y. Clearly [o_{i}, d] = 1 for i = 1, …, r and thus d ∈ C_{G}(C_{W} (T)) = C_{G}(C_{V} (T)) ≤ M, so M = G.

Assume now that C_{G}(C_{V} (T)) ≠ C_{G}(C_{W} (T)). Then there exists i ∈ {1, …, k} and t ∈ T such that [ℓ_{i}, t] ≠ 1. It follows that Λ_{i} ∪ Λ_{i}^{t} = Ω, and {Λ_{i}, Λ_{i}^{t}} is a T -invariant partition of Ω. In particular, every such t acts fixed-point-freely on Ω.

Observe that Λ_{i} ∩ Ω_{j} ≠ ∅ for every j ∈ {1, …, r}; in particular C_{T} (Ω_{j}) ≤ N_{G}(Λ_{i}). If r > 1, then C_{T} (Ω_{2}) is transitive on Ω_{1}, so Ω_{1} ⊆ Λ_{i} and consequently T ≤ N_{G}(Λ_{i}), which contradicts t ∉ N_{G}(Λ_{i}).

We have shown that T is transitive on Ω, so [ℓ_{i}, T] ≠ 1 for every i ∈ {1, …, k}. Let y ∈ T be a 4-cycle acting transitively on Ω_{0} ⊆ Ω. As n ≥ 5, y has a fixed point in Ω and thus y ∈ N_{G}(Λ_{i}) (for every i). In particular either

Ω_{0} ⊆ Λ_{i} or Ω_{0} ⊆ Ω \ Λ_{i}.

In both cases for every i ∈ {1, …, k}

S_{4} ≅ L := N_{G}(Ω_{0}) ∩ C_{G}(Ω \ Ω_{0}) ≤ N_{G}(Λ_{i}) ≤ C_{G}(ℓ_{i}).

It follows that L ≤ C_{G}(C_{V} (T)), but L contains transpositions that are not in Y. Again M = G. □

Lemma 2.12 Let G = S_{n}, n odd, and V be a natural GF (2)G-module. Then the following hold:

(a) C_{V} (Y) = [V, Y].

(b) C_{G}(C_{V} (Y)) = Y.

(c) Let t and t′ be involutions in T. Then t = t′ or C_{V} (t) ≠ C_{V} (t′).

(d) Let d ∈ G with d^{3} = 1 and |[V, d]| = 4. Then d is conjugate to (123) in G.

(e) If G is a minimal parabolic (with respect to 2), then n = 2^{m} + 1.

Proof. Properties (a) – (c) are elementary consequences of the action of G on V^{∗} and Ω.

(d): Let v ∈ Ω such that [v, d] ≠ 1. Then [V^{∗}, d] ≤ ⟨v, v^{d}, v^{d^{−1}}⟩, so d fixes all but 3 elements in Ω. Hence d is conjugate to (123) in G.

(e): We may assume that n ≥ 5, so by 2.11 n is odd. Let M be the unique maximal subgroup containing T. As n is odd, M ≅ S_{n−1} and M has a unique fixed point v ∈ Ω.

Let Ω_{1}, …, Ω_{r} be the T -orbits on Ω with Ω_{1} = {v}. Then T ≤ N_{G}(Ω \ Ω_{2}) ≤ M, so Ω \ Ω_{2} = {v}, and (e) follows. □

Lemma 2.13 Let G = S_{n}, n odd, T ∈ Syl_{2}(G), and U be a GF (2)S_{n}-module. Suppose that U = [U, O^{2}(G)]C_{U}(T) and that [U, O^{2}(G)]/C_{[U,O^{2}(G)]}(O^{2}(G)) is a natural GF (2)S_{n}- module. Then

U = C_{U}(O^{2}(G)) × [U, O^{2}(G)],

in particular [U, O^{2}(G)] is a natural GF (2)S_{n}-module.

Proof. Let U_{0} := C_{U}(O^{2}(G)). It is well known that S_{n} is generated by n − 1 transpositions t_{1}, …, t_{n−1} and it follows from the hypothesis that each of them acts as a transvection on U/U_{0}, so |U/U_{0}| ≤ 2^{n−1}. As the natural GF (2)S_{n}-module has order 2^{n−1}, we conclude that U = [U, O^{2}(G)]U_{0}. Without loss of generality we may assume that |U_{0}| = 2.

It suffices to show that |[U, t_{i}]| = 2, since then ⟨[U, t_{i}] | i = 1, …, n − 1⟩ = [U, G] is a GF (2)S_{n}-submodule of order at most 2^{n−1}, and as above [U, G] has to be a natural GF (2)S_{n}-module.

Let c be an (n − 2)-cycle in A_{n}. Then c is centralized by a transposition t. It is easy to calculate in the natural module that |C_{U/U_{0}}(c)| = 4, so

|C_{U}(c)| = 8 and U = C_{U}(c) × [U, c].

Then [C_{U}(c)/C_{U}(c) ∩ C_{U}(t)| = 2 and |[C_{U}(c), t]| = 2. Moreover, t centralizes [U/U_{0}, c] and thus also [U, c]. It follows that |[U, t]| = 2. □

Lemma 2.14 Let G = S_{n} and V be a natural GF (2)S_{n}-module for G, and let F ≤ G such that F = O^{2}(F) and [V, F]C_{V} (F)/C_{V} (F) is an irreducible GF (2)F -module. Then the following hold:

(a) F has a unique non-trivial orbit on Ω.

(b) Suppose that n is odd, F ≅ A_{k}, k odd, and [V, F] is a natural A_{k}-module for F. Then F is normalized by a conjugate of Y.

(c) Suppose that F ≅ SL_{2}(2^{k}) and [V, F]/C_{[V,F]}(F) is a natural SL_{2}(2^{k})-module for F. Then k = 2, and F has exactly n − 6 fixed-points on Ω. In particular [V, F] and C_{[V,F]}(F) are normalized by a conjugate of Y.

Proof. Observe that C_{V} (F) = C_{V^{∗}}(F)/C_{V^{∗}}(G) since F = O^{2}(F), so

[V^{∗}, F]C_{V^{∗}}(F)/C_{V^{∗}}(F) ≅ [V, F]C_{V} (F)/C_{V} (F) ≅ [V, F]/C_{[V,F]}(F).

(a): For v ∈ Ω, let W_{v} := ⟨v^{F} ⟩. As Ω is a basis of V^{∗}, we get for v, ṽ ∈ Ω

W_{v} = Wṽ and v^{F} = ṽ^{F} or W_{v} ∩ Wṽ = 1.

Now the irreducibility of [V^{∗}, F]C_{V^{∗}}(F)/C_{V^{∗}}(F) shows that [W_{v}, F] = 1 for all but one orbit v^{F}.

(b): According to (a) F has a unique non-trivial orbit Ω_{0} ⊆ Ω. Set m := |Ω_{0}| and W_{0} := ⟨Ω_{0}⟩. Then |W_{0}| = 2^{m} and |[W_{0}, F]| = 2^{m−1}. As [W_{0}, F] is also a natural A_{k}-module for F we also get that |[W_{0}, F]| = 2^{k−1}, so k = m. Moreover, since k and n are odd, |Ω \ Ω_{0}| is even. Hence, there exists a conjugate of Y normalizing Ω \ Ω_{0} and thus also F.

(c): As in the proof of (b) we define W_{0} using the unique non-trivial orbit Ω_{0} of F on Ω and set m := |Ω_{0}|. Observe that C_{W_{0}}(F) = ⟨∏_{w∈Ω_{0}} w⟩ and that [W_{0}, F] is the set of all products of an even number of elements of Ω_{0}. On the other hand [W_{0}, F]C_{W_{0}}(F)/C_{W_{0}}(F) is a natural SL_{2}(2^{k})-module for F, so F is transitive on the non-trivial elements of [W_{0}, F]C_{W_{0}}(F)/C_{W_{0}}(F). It follows that every element of [W_{0}, F]\ C_{W_{0}}(F) is either the product of m−2 or 2 elements of Ω_{0}. Since |F | ≥ 60 we get m ≥ 5 and 4 = m − 2, so m = 6. In particular F is a subgroup of A_{6} and thus k = 2.

We have that [W_{0}, F] = [W_{0}, C_{G}(Ω \ Ω_{0})] and C_{[W_{0},F]}(F) = C_{[W_{0},F]}(C_{G}(Ω \ Ω_{0})). As there exists a conjugate of Y normalizing Ω_{0} and Ω\Ω_{0}, this conjugate also normalizes [W_{0}, F] and C_{[W_{0},F]}(F). Now the additional statement in (c) follows.

□

Lemma 2.15 Let G = S_{n}, n ≥ 3 and n odd, and let V be a natural GF (2)S_{n}-module for G. Suppose that A ∈ O_{G}(V). Then the following hold:

(a) A is generated by commuting transpositions of G.

(b) [V, A, A] = 1.

(c) |V/C_{V} (A)| = |A|.

Proof. We proceed by induction on n. The case n = 3 is trivial, so we assume that n ≥ 5 and that the result holds for n − 2. Since V^{∗} = V × V_{0} we may as well calculate in V^{∗} rather than V.

By the Timmesfeld Replacement Theorem [12] there exists 1 ≠ A_{0} ≤ A such that [V^{∗}, A_{0}, A] = 1 and A_{0} ∈ O_{G}(V^{∗}). Let 1 ≠ a ∈ A_{0} and v ∈ Ω such that v ≠ v^{a} and let t be the transposition of G with v^{t} = v^{a} and V_{t}^{∗} := C_{V^{∗}}(t). Then w := vv^{a} = vv^{t} ∈ C_{V^{∗}}(A), so

A ≤ C_{G}(w) = ⟨t⟩ × L, L ≅ S_{n−2}.

Observe that V_{t}^{∗}/⟨w⟩ is the natural permutation module for L. Thus by induction and 2.8 |A/C_{A}(V_{t}^{∗})| = |V_{t}^{∗}/C_{V_{t}^{∗}}(A)|, and A = ⟨t_{1}, …, t_{r}⟩C_{A}(V_{t}^{∗}), where t_{1}, …, t_{r} are commuting transpositions of G in L. Moreover, by 2.12 C_{A}(V_{t}^{∗}) ≤ ⟨t⟩, so (a) and (c) follow, and 2.12 (a) yields (b). □

Lemma 2.16 Let V be a finite dimensional GF (p)G-module, E ⊴⊴ G, and W := [V, E], and let A ∈ O_{G}(V) with [E, A] ≠ 1. Suppose that

(i) E ≅ SL_{2}(p^{m})′ or p = 2 and E ≅ A_{2m+1}, and

(ii) W/C_{W} (E) is a natural SL_{2}(p^{m})′- resp. A_{2m+1}-module for E.

Then the following hold:

(a) A ≤ N_{G}(E).

(b) \overline{EA} := EA/C_{EA}(W) ≅ SL_{2}(p^{m}) and \overline{A} ∈ Syl_{p}(\overline{EA}), or p = 2, \overline{EA} ≅ S_{2m+1} and \overline{A} is generated by commuting transpositions.

(c) [W, A, A] = 1.

(d) |A/C_{A}(W)| = |W/C_{W} (A)|.

(e) For T ∈ Syl_{p}(EA) there exists a unique maximal element B in O_{T} (V), and C_{EA}(C_{V} (B)) = B.

Proof. We may assume that G = ⟨E, A⟩. Let A_{0} := C_{A}(E), A = A_{0} × A_{1}, and V_{0} := C_{V} (A_{0}). The P × Q-Lemma shows that E acts faithfully on V_{0}. Moreover, W C_{V} (A) ≤ V_{0} since W = [W, E] and W/C_{W} (E) is an irreducible E-module. In addition, by 2.8 (c) A_{1} ∈ O_{G}(V_{0}), so A_{1}, V_{0} and E satisfy the hypothesis in place of A, V and E. Hence, we may assume A_{0} = 1 and V = V_{0}.

(a): This follows from [4] if E is quasi-simple and from [12, 9.3.6] if E is solvable.

(b) – (e): Suppose first that E ≅ A_{2m+1}. By (a) and 2.15 \overline{EA} ≅ S_{2m+1} and W/C_{W} (E) is a natural S_{2m+1}-module. Now 2.13 yields C_{W} (E) = 1, and (b) – (d) follow. Moreover, again by 2.15, a maximal element B ∈ O_{T} (V) is generated by a set which corresponds to a maximal set of pairwise commuting transpositions in S_{2m+1}, so B is unique and 2.12 yields (e).

Suppose now that E ≅ SL_{2}(p^{m})′. As one can see in Aut(SL_{2}(p^{m})), |A| ≤ p^{m} since A is abelian, so |W/C_{W} (A)| ≤ p^{m}. On the other hand, A induces a group of semi-linear GF (p^{m})-transformations on W/C_{W} (E). It follows that |W/C_{W} (A)| = |A| = p^{m} and EA ≅ SL_{2}(p^{m}). Now (b) – (e) are easy to verify. □

3 Minimal Parabolic Groups

Throughout this section we assume

Hypothesis 3.1 Let P be a minimal parabolic group with respect to p, T ∈ Syl_{p}(P), and let M be the unique maximal subgroup of P containing T.

Lemma 3.2 (L-Lemma) Let A ≤ T with A ≰ O_{p}(P). Then there exists a subgroup L containing A such that the following hold:

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

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

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

Proof. See [15].

□

Lemma 3.3 Suppose N ⊴ P. Then the following hold:

(a) If N ≤ M, then N ∩ T ◁ P.

(b) If N ≰ M, then O^{p}(P) ≤ N.

Proof. See [15, 1.3(b)]. □

Lemma 3.4 Let N be a normal subgroup of P contained in M. Set \overline{P} := P/N. Then \overline{P} is a minimal parabolic group and O_{p}(\overline{P}) = \overline{O_{p}(P)}.

Proof. Observe that \overline{T} ∈ Syl_{p}(\overline{P}) and \overline{M} is the unique maximal subgroup of \overline{P} containing \overline{T}. Suppose that \overline{T} ◁ \overline{P}. Then T N is a normal subgroup of P. Since T N ≤ M, Lemma 3.3 (a) gives T = T N ∩ T ◁ P, which contradicts the assumption that P is minimal parabolic. Therefore \overline{P} is a minimal parabolic group.

Let D be the inverse image of O_{p}(\overline{P}) in P. Then D ⊴ P. Since D ≤ T N ≤ M, by Lemma 3.3 (a), we have that D ∩ T ◁ P. Then using the Dedekind Identity, D = (D ∩ T)N ≤ O_{p}(P)N. Hence O_{p}(\overline{P}) = \overline{D} ≤ \overline{O_{p}(P)}. The reverse inclusion always holds, so O_{p}(\overline{P}) = \overline{O_{p}(P)}. □

Lemma 3.5 Let V be a faithful GF (p)-module for P. Suppose that there exists an elementary abelian subgroup A ≤ T such that:

(i) |V/C_{V} (A)| ≤ |A| and |A_{0}||C_{V} (A_{0})| < |A||C_{V} (A)| for every 1 ≠ A_{0} < A,

(ii) [C_{V} (T), P] ≠ 1, and

(iii) P = ⟨A, A^{x}⟩ for every x ∈ P \ M.

Then P ≅ SL_{2}(q), q := |A|, C_{V} (A) = [V, A]C_{V} (P), and V/C_{V} (P) is a natural SL_{2}(q)- module for P.

Proof. We will use the following additional notation:

Z := C_{V} (T), W := ⟨Z^{P} ⟩, Ṽ := V/C_{V} (P), \overline{P} := P/C_{P} (W).

3.5.1 A acts quadratically on V and [W, A] ≠ 1.

The first part follows from [12, 9.2.1] together with (i) and the second part follows from (ii) and (iii).

3.5.2 O_{p}(\overline{P}) = C\overline{P} (W̃) = 1 and \overline{M} is a maximal subgroup of \overline{P}.

Note that C_{P} (W̃)/C_{P} (W) is a p-group, so C_{P} (W̃) ≤ O_{p}(\overline{P}). Let C be the inverse image of O_{p}(\overline{P}). Then 3.3 implies that

C_{P} (W)T = P or C ≤ C_{P} (W)O_{p}(P) ≤ M.

In the first case P = C_{P} (Z), which contradicts (ii). In the second case C = C_{P} (W), since O_{p}(P) ≤ C_{P} (W), so O_{p}(\overline{P}) = 1. Moreover, \overline{M} is a maximal subgroup of \overline{P}, since C ≤ M.

3.5.3 CW̃(P) = 1 and { \atop C_{W} (A)} = CW̃(A).

Let x ∈ P \ M and put B := A^{x}, so P = ⟨A, B⟩ by (iii). The quadratic action of A implies that

W = [W, A][W, B]Z ≤ C_{W} (A)C_{W} (B) ≤ W,

and we must have equality. Therefore

W̃ = { \atop C_{W} (A)}{ \atop C_{W} (B)} and { \atop C_{W} (A)} ∩ { \atop C_{W} (B)} = { \atop C_{W} (P)} = 1.

As A and B are conjugate in P, we also get that

CW̃(A) ∩ { \atop C_{W} (B)} = 1, and thus { \atop C_{W} (A)} = CW̃(A).

Now CW̃(P) = 1 follows.

3.5.4 |W̃/CW̃(\overline{A})| ≤ |\overline{A}|.

Let A_{0} := C_{A}(W̃). By 3.5.2 [A_{0}, W] = 1. Hence (i) gives

(1)  |A_{0}||W C_{V} (A)| ≤ |A_{0}||C_{V} (A_{0})| ≤ |A||C_{V} (A)|.

This shows that

(2)  |W̃/CW̃(A)| ≤ |W/C_{W} (A)| ≤ |A/A_{0}|.

3.5.5 There exists a field K with |K| = |\overline{A}| such that W̃ is a 2-dimensional vector space over K and \overline{P} = SL(W̃, K).

According to 3.5.1 – 3.5.4 and (iii), \overline{P} satisfies the hypothesis of [7, Theorem 2] and 3.5.5 follows from this theorem.

From 3.5.3 and 3.5.5 we get that

|W/C_{W} (A)| = |\overline{A}| and C_{W} (A) = [W, A]C_{W} (P).

Hence (1) and (2) give

|A_{0}||W C_{V} (A)| = |A_{0}||C_{V} (A_{0})| = |A||C_{V} (A)|,

so by (i), A_{0} = 1, |\overline{A}| = |A|, and |V/C_{V} (A)| = |A|. From (iii) we get that V = W C_{V} (P) and then V = W since C_{V} (P) ≤ C_{V} (T) ≤ W. In particular C_{P} (W) = C_{P} (V) = 1, so \overline{P} = P. □

Theorem 3.6 Let V be a faithful GF (p)-module for P. Suppose that O_{p}(P) = 1, A_{P} (V) ≠ ∅, and C_{P} (C_{V} (T)) ≤ M. Then for every A ∈ A_{P} (V) there exists a subgroup L_{0} ≤ P with A ≤ L_{0} such that the following hold:

(a) [V, A, A] = 1.

(b) L_{0} ≅ SL_{2}(q), q := |A|, V/C_{V} (L_{0}) is a natural SL_{2}(q)-module for L_{0}, and C_{V} (A) = [V, A]C_{V} (L_{0}); in particular |V/C_{V} (A)| = |A|.

(c) C_{V} (A) = C_{V} (a) for every a ∈ A^{♯}.

(d) |V/C_{V} (AB)| = |A||B| for every B ∈ A_{P} (V) \ {A} with [A, B] = 1.

Proof. Let A ∈ A_{P} (V). Then the maximality of |A||C_{V} (A)| and minimality of A give

3.6.1 |V/C_{V} (A)| ≤ |A|, and |A_{0}||C_{V} (A_{0})| < |A||C_{V} (A)| for every 1 ≠ A_{0} < A.

We now apply the L-Lemma 3.2. Then there exists A ≤ L ≤ P and g ∈ P such that

3.6.2 L ∩ M^{g} is the unique maximal subgroup of L containing AO_{p}(L).

3.6.3 L = ⟨A, A^{x}⟩O_{p}(L) for every x ∈ L \ M^{g}.

3.6.4 L is not contained in any P -conjugate of M.

Among all x ∈ L \ M^{g} we choose B := A^{x} such that L_{0} := ⟨A, B⟩ is minimal. We prove next:

3.6.5 L_{0} is minimal parabolic, and L_{0} and V satisfy the hypothesis of 3.5.

The first part of 3.6.5 follows from the fact that L is minimal parabolic by 3.6.3 and that L = L_{0}O_{p}(L). Hypothesis (i) of 3.5 follows from 3.6.1 and Hypothesis (iii) follows from the definition of L_{0}. Let T_{0} ∈ Syl_{p}(L_{0}) with T_{0} ≤ T and suppose [L_{0}, C_{V} (T_{0})] = 1. Then

L = O_{p}(L)L_{0} ≤ O_{p}(L)C_{L}(C_{V} (T_{0})) ≤ C_{L}(C_{V} (T)) ≤ L ∩ M,

which contradicts 3.6.4. Thus Hypothesis (ii) of 3.5 holds.

Now properties (a) – (c) follow from 3.5 and elementary properties of the natural SL_{2}(q)-module.

For the proof of (d), let B ∈ A_{P} (V) such that [A, B] = 1. If A ∩ B ≠ 1, then by (iii), C_{V} (A) = C_{V} (B) and the maximality of |A||C_{V} (A)| shows that A = B. If A ∩ B = 1, then the maximality of |A||C_{V} (A)| = |B||C_{V} (B)| = |V | gives

|AB| ≤ |V/C_{V} (AB)| = |V/C_{V} (A) ∩ C_{V} (B)| ≤ |V/C_{V} (A)||V/C_{V} (B)| = |A||B| = |AB|.

□

Lemma 3.7 Let P be of characteristic p and W := Ω(Z(O_{p}(P))). Suppose that neither Ω(Z(T)) nor B(T) is normal in P and that P/C_{P} (W) ≅ SL_{2}(p^{n}). Then B(T) ∈ Syl_{p}(⟨B(T)^{P} ⟩) and Ω(Z(B(T)))W ⊴ P.

Proof. See [15, 2.7]. □

Lemma 3.8 Let G be of characteristic p, C^{∗}(G, T) ≠ G for T ∈ Syl_{p}(G), and V ⊴ G with

Ω(Z(T)) ≤ V ≤ Ω(Z(O_{p}(G))).

Suppose that G/C_{G}(V) ≅ SL_{2}(p^{n}) or S_{2m+1} (with p = 2) and V/C_{V} (G) is a natural SL_{2}(p^{n})- resp. S_{2m+1}-module for G/C_{G}(V). Then there exists a B(T)-block E of G such that G = B(T)EC_{G}(V) and [E, Ω(Z(B(T)))] ≤ V.

Proof. Let E := O^{p}(G). Assume first that G/C_{G}(V) ≅ SL_{2}(p^{n}). Clearly C_{G}(V) ≤ C^{∗}(G, T) and with the Frattini argument B(T) ≰ C_{G}(V). Then B(T)C_{G}(V) = T C_{G}(V), and N_{G}(T)C_{G}(V) is the unique a maximal subgroup of G that contains B(T)C_{G}(V); in particular C^{∗}(G, T) = N_{G}(B(T))C_{G}(V). By 2.3 (e) there exists a minimal parabolic subgroup P of characteristic p in G such that

B(T) ≤ T_{0} ∈ Syl_{p}(P), P ≠ C^{∗}(P, T_{0}), and P ≰ C^{∗}(G, T).

Thus P C_{G}(V) ≰ N_{G}(B(T))C_{G}(V) and P C_{G}(V) = G. So we may assume without loss that P = G and by 3.7 that also B(T) = T_{0} and [O^{p}(P), Ω(Z(B(T)))] ≤ V. In particular, no non-trivial characteristic subgroup of T_{0} is normal in P. Now a standard pushing up result, see for example [16], shows that O^{p}(P) is a B(T)-block and the result holds with E := O^{p}(P).

Assume now that p = 2 and \overline{G} := G/C_{G}(V) ≅ S_{2m+1}. Again by the Frattini argument B(T) ≰ C_{G}(V), so 2.8 (e) yields A ∈ O_{G}(V) with [O^{2}(G), A] ≠ 1. Then by 2.16 \overline{B(T)} is generated by a maximal set of pairwise commuting transpositions \overline{t}_{1}, …, \overline{t}_{m}. Since 2m + 1 is odd, for every i there exists a 3-cycle \overline{d}_{i} such that [\overline{d}_{i}, \overline{t}_{j}] = 1 for i ≠ j and

⟨\overline{d}_{i}, \overline{t}_{i}⟩ ≅ S_{3} ≅ SL_{2}(2).

Let L_{i} be the inverse image of \overline{B(T)}⟨\overline{d}_{i}, \overline{t}_{i}⟩ in G and G_{0} := ⟨L_{1}, …, L_{m}⟩. Then G = G_{0}C_{G}(V); in particular L_{i} ≰ C^{∗}(G, T) for i = 1, …, m.

Now 2.3 shows that L_{i} satisfies the hypothesis with L_{i}/C_{L_{i}}([V, L_{i}]) ≅ SL_{2}(2). Hence, there exists a B(T)-block E_{i} ≤ L_{i} and [Ω(Z(B(T))), E_{i}] ≤ V. Let E = ⟨E_{1}, …, E_{m}⟩. Then [E, O_{2}(G)Ω(Z(B(T)))] ≤ V and thus C_{E}(V) ≤ O_{2}(G) since G is of characteristic 2. It follows that E is a B(T)-block with E/O_{2}(E) ≅ A_{2m+1} and G = B(T)EC_{G}(V). □

4 Conjugacy Classes of Transvections

In this section we will work with the following hypotheses: Hypothesis 4.1 Let P be a group acting faithfully on an elementary abelian p-group V. Suppose that there exists a normal set^{1} D of non-trivial elementary abelian psubgroups of P such that the following hold for A ∈ D :

(i) [V, A, A] = 1. (ii) |V/C_{V} (A)| = |A| and C_{V} (A) = C_{V} (a) for every a ∈ A^{♯}. (iii) |V/C_{V} (AB)| = |A||B| for every B ∈ D with B ≠ A and [A, B] = 1. For U ≤ P we set

D ∩ U := {A | A ∈ D, A ≤ U} and D_{P} (U) := ∩_{g∈P} (D ∩ U^{g}).

Hypothesis 4.2 Assume Hypothesis 4.1 and, in addition, that T ∈ Syl_{p}(P) and T ≤ M ≤ P with D ≠ D_{P} (M) such that (∗) N_{P} (D ∩ T) ≤ M and C_{P} (C_{V} (T)) ≤ M. Hypothesis 4.3 Assume Hypothesis 4.2 and in addition that (∗∗) |A||C_{V} (A)| ≥ |X||C_{V} (X)| for every A ∈ D and every elementary abelian p-

subgroup X ≤ P. Notation 4.4 Assume Hypothesis 4.2. For A ∈ D we set

M(A) := {M^{g} | g ∈ P, A ≤ M^{g}}.

By Λ we denote the set of all subgroups L ≤ P such that

(1) L ≅ SL_{2}(q) and V/C_{V} (L) is a natural SL_{2}(q)-module for L,

(2) D ∩ L is the set of Sylow p-subgroups of L,

(3) M(A) ≠ M(B) for A ≠ B ∈ D ∩ L. Moreover Λ(A) := {L ∈ Λ | A ≤ L}. Lemma 4.5 Assume Hypothesis 4.1. Let A, B ∈ D. Then A = B or A ∩ B = 1.

1i.e., invariant under conjugation by G.

Proof. Let x ∈ A ∩ B. Suppose that x ≠ 1. By 4.1(ii)

C_{V} (A) = C_{V} (x) = C_{V} (B).

Now 4.1(i) gives [V, A, B] ≤ [C_{V} (A), B] = 1 and similarly [B, V, A] = 1, so the Three Subgroups Lemma yields [A, B, V] = 1. Therefore [A, B] = 1, because P acts faithfully on V. Thus 4.1(iii) gives the result. □

Lemma 4.6 Assume Hypothesis 4.1. Let A, B ∈ D such that A ≠ B and [A, B] = 1. Then V = C_{V} (A)C_{V} (B) and AB acts quadratically on V.

Proof. We have

|A||B||C_{V} (AB)| {4.5 \atop =} |AB||C_{V} (AB)| {4.1 \atop =} |A||C_{V} (A)|.

Hence

|B| = |C_{V} (A)/C_{V} (AB)| = |C_{V} (A)C_{V} (B)/C_{V} (B)| ≤ |V/C_{V} (B)| {4.1 \atop =} |B|,

and thus V = C_{V} (A)C_{V} (B). In particular

[V, A] = [C_{V} (B), A] ≤ C_{V} (B) ∩ C_{V} (A)

and similarly [V, B] ≤ C_{V} (A) ∩ C_{V} (B). □

Lemma 4.7 Assume Hypothesis 4.1. Then ⟨D∩T ⟩ is elementary abelian, and ⟨D∩T ⟩ acts quadratically on V.

Proof. If ⟨D ∩ T ⟩ is abelian, then by 4.6 it also acts quadratically on V. Thus, it suffices to show that ⟨D ∩ T ⟩ is abelian.

Suppose on the contrary that ⟨D ∩ T ⟩ is not elementary abelian. Then there exist A_{1}, A_{2} ∈ D ∩ T with [A_{1}, A_{2}] ≠ 1; in particular A_{1} ≠ A_{2}. Choose ⟨A_{1}, A_{2}⟩ minimal with this property.

Since a p-group cannot be generated by conjugates of a proper subgroup, we have

4.7.1 ⟨A_{1}^{A_{2}}⟩ ≠ ⟨A_{1}, A_{2}⟩ ≠ ⟨A_{2}^{A_{1}}⟩.

Then by the minimality of ⟨A_{1}, A_{2}⟩:

4.7.2 ⟨A_{1}^{A_{2}}⟩ and ⟨A_{2}^{A_{1}}⟩ are elementary abelian.

If A_{1} ≤ N_{T} (A_{2}) and A_{2} ≤ N_{T} (A_{1}) then by 4.5

[A_{1}, A_{2}] ≤ A_{1} ∩ A_{2} = 1,

and ⟨A_{1}, A_{2}⟩ is elementary abelian, which is a contradiction. Thus we may assume without loss that A_{2} ≰ N_{T} (A_{1}).

Pick a ∈ A_{2} \ N_{T} (A_{1}). Then 4.7.2 and 4.6 show that

V = C_{V} (A_{1})C_{V} (A_{1}^{a}) = C_{V} (A_{1})C_{V} (A_{1})^{a} = C_{V} (A_{1})[V, a].

Since A_{2} acts quadratically on V, we get

V = C_{V} (A_{1})C_{V} (A_{1}^{a}) = C_{V} (A_{1})C_{V} (A_{2}).

Observe that C_{V} (A_{1}) ∩ C_{V} (A_{2}) ≤ C_{V} (A_{1}) ∩ C_{V} (A_{1}^{a}). So 4.1 gives

4.7.3 |A_{2}| = |V/C_{V} (A_{2})| = \frac{|C_{V} (A_{1})|}{|C_{V} (A_{1})∩C_{V} (A_{2})|} ≥ \frac{|C_{V} (A_{1})|}{|C_{V} (A_{1})∩C_{V} (A_{1}^{a})|} = |V/C_{V} (A_{1}^{a})| = |A_{1}|.

If also A_{1} ≰ N_{T} (A_{2}), then a symmetric argument shows |A_{1}| ≤ |A_{2}|, so |A_{1}| = |A_{2}|. If A_{1} ≤ N_{T} (A_{2}), then A_{1}A_{1}^{a} ∩ A_{2} ≠ 1 and by 4.1

C_{V} (A_{1})∩C_{V} (A_{2}) ≤ C_{V} (A_{1})∩C_{V} (A_{1}^{a}) ≤ C_{V} (A_{1})∩C_{V} (A_{1}A_{1}^{a} ∩A_{2}) = C_{V} (A_{1})∩C_{V} (A_{2}),

so C_{V} (A_{1}) ∩ C_{V} (A_{2}) = C_{V} (A_{1}) ∩ C_{V} (A_{1}^{a}). This gives equality in 4.7.3 and again |A_{1}| = |A_{2}|. But then A_{1}A_{1}^{a} = A_{1}A_{2}, which contradicts 4.7.1. We have shown:

4.7.4 |A_{1}| = |A_{2}| and also A_{1} ≰ N_{T} (A_{2}).

Pick b ∈ A_{1} \ N_{T} (A_{2}). By 4.1 and 4.7.4

|V/C_{V} (A_{1}) ∩ C_{V} (A_{2})| ≤ |A_{1}||A_{2}| = |A_{1}|^{2} = |V/C_{V} (A_{1}) ∩ C_{V} (A_{1}^{a})|,

This gives C_{V} (A_{1}) ∩ C_{V} (A_{2}) = C_{V} (A_{1}) ∩ C_{V} (A_{1}^{a}) and with a symmetric argument C_{V} (A_{1}) ∩ C_{V} (A_{2}) = C_{V} (A_{2}) ∩ C_{V} (A_{2}^{b}).

On the other hand, by 4.7.2 and 4.6 both subgroups A_{1}A_{1}^{a} and A_{2}A_{2}^{b} act quadratically on V, so

[V, A_{1}] ≤ C_{V} (A_{1}) ∩ C_{V} (A_{1}^{a}) = C_{V} (A_{1}) ∩ C_{V} (A_{2})

and

[V, A_{2}] ≤ C_{V} (A_{2}) ∩ C_{V} (A_{2}^{b}) = C_{V} (A_{1}) ∩ C_{V} (A_{2}).

It follows that [V, A_{1}, A_{2}] = [V, A_{2}, A_{1}] = 1, and the Three Subgroups Lemma yields [A_{1}, A_{2}, V] = 1. But then [A_{1}, A_{2}] = 1 since P is faithful on V, a contradiction. □

Lemma 4.8 Assume Hypothesis 4.1. Let A, B ∈ D such that [A, B] ≠ 1 and set L := ⟨A, B⟩. Then for every C ∈ D ∩ L with [C, A] = 1 either C ≤ Z(L) or C = A. In particular, for X, Y ∈ D either X and Y are conjugate in ⟨X, Y ⟩, or [X, Y] = 1.

Proof. Let L be a counterexample, so there exists C ∈ D ∩ L such that [C, A] = 1 but C ≠ A and [C, B] ≠ 1.

Assume first that C is conjugate to B. Then |C| = |B|, and 4.1 (iii) implies

|V/C_{V} (AC)| = |A||B|.

On the other hand by 4.1 (ii) |V/C_{V} (L)| ≤ |A||B|, so we get that C_{V} (L) = C_{V} (AC). Now 4.6 shows that [V, A] ≤ C_{V} (L). Hence ⟨A^{L}⟩ acts quadratically on V and A ≤ O_{p}(L). But then by 4.7 [A, B] = 1, a contradiction.

Assume now that C is not conjugate to B. Then there exists a Sylow p-subgroup of L_{0} := ⟨C, B⟩ that contains B and a conjugate C^{∗} of C; in particular by 4.7 [C^{∗}, B] = 1. With the same argument as in the first case, this time applied to L_{0}, we get C_{V} (L_{0}) = C_{V} (C^{∗}B) and then [V, B] ≤ C_{V} (L_{0}), so as above [C, B] = 1, a contradiction.

We have shown that L has the desired properties. Let x ∈ L such that ⟨B^{x}, A⟩ is a p-group. Then 4.7 implies [B^{x}, A] = 1 and thus A = B^{x} since B^{x} ≰ Z(L). Now the second part of the assertion follows. □

Lemma 4.9 Assume Hypothesis 4.2. Let H ≤ P such that D ∩ T ⊆ D ∩ H and D ∩ H ⊈ D ∩ M. Then H satisfies Hypothesis 4.2 with respect to D ∩ H and M ∩ H.

Proof. Let T_{0} ∈ Syl_{p}(H) such that D ∩ T = D ∩ T_{0}. Then N_{H}(D ∩ T_{0}) ≤ M ∩ H; in particular T_{0} ≤ M and T_{0} ≤ T^{g} for some g ∈ M. It follows that

C_{V} (T^{g}) ≤ C_{V} (T_{0}) and C_{H}(C_{V} (T_{0})) ≤ C_{H}(C_{V} (T^{g})) ≤ M ∩ H.
□

Lemma 4.10 Assume Hypothesis 4.2. Let D_{0} ⊆ D be a normal subset of P such that D_{0} ⊈ D_{P} (M). Then ⟨D_{0}⟩ satisfies Hypothesis 4.2 with respect to D_{0} and M ∩ ⟨D_{0}⟩.

Proof. Let D_{1} := D \ D_{0}, P_{0} := ⟨D_{0}⟩, and T_{0} := P_{0} ∩ T. Observe that by 4.8 [P_{0}, ⟨D_{1}⟩] = 1; in particular

D ∩ T = (D_{0} ∩ T_{0}) ∪ C_{D∩T} (P_{0}).

It follows that

N_{P_{0}}(D_{0} ∩ T_{0}) ≤ N_{P_{0}}(D ∩ T) ≤ M ∩ P_{0}.

As also

C_{P_{0}}(C_{V} (T_{0})) ≤ C_{P_{0}}(C_{V} (T)) ≤ M ∩ P_{0},

the claim now follows from the fact that D_{0} ⊈ D_{P} (M). □

Lemma 4.11 Assume Hypothesis 4.2. Let T_{0} ⊆ D ∩ T be maximal (by inclusion) such that N := N_{P} (T_{0}) ≰ M. Then

D ∩ N ≠ D ∩ N ∩ M and D ∩ N ∩ M ≠ T_{0},

and ⟨A, B⟩ ∈ Λ for every A ∈ (D ∩ M ∩ N) \ T_{0} and B ∈ (D ∩ N) \ (D ∩ M).

Proof. Set T := D ∩ T. Recall from 4.7 that the elements in T centralize each other, and from 4.2 that N_{P} (T) ≤ N_{P} (T ) ≤ M. The Frattini argument shows that the only P -conjugate of M containing T is M itself. Let T_{1} ⊆ T . As T ⊆ N_{P} (T_{1}), an elementary argument gives

4.11.1 Either N_{P} (T_{1}) ≰ M, or M is the unique conjugate of M containing T_{1}.

In particular D ∩ N ≠ D ∩ N ∩ M, and D ∩ N ∩ M ≠ T_{0}. Let

A ∈ (D ∩ N ∩ M) \ T_{0}, B ∈ (D ∩ N) \ (D ∩ M) and L := ⟨A, B⟩

such that L is a minimal counterexample. We also set

D^{∗} := A^{L} and H := L ∩ M.

As N_{L}(A) ≤ N_{P} (T_{0} ∪ {A}), the maximality of T_{0} and 4.11.1 imply:

4.11.2 H is the unique L-conjugate of H containing A; in addition

N_{L}(A) ≤ H, N_{L}(H) = H and [A, B] ≠ 1.

By 4.8 A and B are conjugate in L, so q := |A| = |B|. We now divide the proof into two cases.

4.11.3 Case I : There exists X ∈ D^{∗} such that L_{0} := ⟨A, X⟩ < L and X ≰ H.

The minimality of L shows that L_{0} ∈ Λ; in particular L_{0} ≅ SL_{2}(q) and V/C_{V} (L_{0}) is a natural SL_{2}(q)-module for L_{0}. By 4.1 (ii) |V/C_{V} (L)| ≤ |A||B| = q^{2} while |V/C_{V} (L_{0})| = q^{2}. Since C_{V} (L_{0}) ≥ C_{V} (L) we get that C_{V} (L) = C_{V} (L_{0}).

Let A_{0}, …, A_{q} be the Sylow p-subgroups of L_{0} with A_{0} := A. As V/C_{V} (L_{0}) is a natural SL_{2}(q)-module, the groups C_{V} (A_{i})/C_{V} (L_{0}), i = 0, …, q, form a partition of V/C_{V} (L_{0}). Thus, there exists i ∈ {0, …, q} such that

C_{V} (L) = C_{V} (L_{0}) < C_{V} (B) ∩ C_{V} (A_{i}).

Let L_{i} := ⟨A_{i}, B⟩. Then C_{V} (L) < C_{V} (L_{i}), so L_{i} < L. The minimality of L shows that either V/C_{V} (L_{i}) is a natural SL_{2}(q)-module, or L_{i} ≤ H^{x} where x ∈ L with B ≤ H^{x}.

The first case contradicts |V/C_{V} (L_{i})| < |V/C_{V} (L)| = q^{2}. In the second case i is uniquely determined since any two different Sylow p-subgroups generate L_{0} and A ≰ H^{x} by 4.11.2. It follows that C_{V} (A_{i}) = C_{V} (B); in particular [A_{i}, B] = 1. Hence 4.1 (iii) yields A_{i} = B and L = L_{0}. But then L is not a counterexample.

4.11.4 Case II : X ≤ H for every X ∈ D^{∗} with ⟨A, X⟩ < L.

Let T_{0} ∈ Syl_{p}(L) with A ≤ T_{0}, and let x ∈ L \ H. By 4.11.2 A^{x} ≤ H implies x ∈ H. As we are in Case II, this shows that

L = ⟨A, A^{x}⟩ for every x ∈ L \ H.

By 4.11.2 T_{0} ≤ T^{h} ≤ M, for some h ∈ H, so C_{V} (T^{h}) ≤ C_{V} (T_{0}), and thus by 4.2

C_{L}(C_{V} (T_{0})) ≤ C_{L}(C_{V} (T^{h})) ≤ H.

Now (A, L, H) satisfies the hypothesis of 3.5 in place of (A, P, M) and L is not a counterexample. □

Lemma 4.12 Assume Hypothesis 4.2. For every A ∈ D \ D_{P} (M) there exists g ∈ P and L ∈ Λ(A) such that A ≤ M^{g} and L ≰ M^{g}. In particular Λ(A) ≠ ∅.

Proof. Let D_{0} be the set of all A ∈ D such that there exists a g ∈ P and L ∈ Λ(A) such that A ≤ M^{g} and L ≰ M^{g}. We set

D^{∗} := D_{0} ∪ D_{P} (M) and D_{∗} := D \ D^{∗}.

We have to show that D = D^{∗}.

Observe that D^{∗} and D_{∗} are normal sets in P, so no element of D_{∗} is conjugate to an element of D^{∗}. Hence 4.8 shows that the elements of D_{∗} centralize the elements of D^{∗}.

From now on we assume that D_{∗} ≠ ∅ and derive a contradiction. Let T_{1} := D^{∗} ∩ T. Then D_{∗} ⊆ N_{P} (T_{1}), so N_{P} (T_{1}) ≰ M, since D_{∗} is a normal set. We now choose T_{0} ⊆ D ∩ T maximal with respect to T_{1} ⊆ T_{0} and N_{P} (T_{0}) ≰ M. Observe that D^{∗} ∩ N_{P} (T_{0}) = D^{∗} ∩ T = T_{1}.

According to 4.11 there exist A ∈ (D ∩ M ∩ N_{P} (T_{0})) \ T_{0} and L ∈ Λ(A) such that L ≤ N_{P} (T_{0}) and L ≰ M; in particular A ∈ D^{∗}. It follows that A ∈ D^{∗} ∩ N_{P} (T_{0}) = T_{1} ⊆ T_{0}, a contradiction.

□

Lemma 4.13 Assume Hypothesis 4.2. Let L ∈ Λ and B ∈ D such that [L, B] ≠ 1 and B ≰ L. Then there exists a unique A ∈ D ∩ L such that the following hold for L^{∗} := ⟨L, B⟩, q := |A| and \overline{V} := V/C_{V} (L^{∗}) :

(a) [A, B] = 1,

(b) C_{V} (L^{∗})[V, A] = C_{V} (L^{∗})[V, B] = C_{V} (AB),

(c) |[\overline{V} , A]| = q,

(d) |\overline{V} | = q^{3}, and

(e) [\overline{V} , L] = [\overline{V} , L^{∗}] is a natural SL_{2}(q)-module for L invariant under L^{∗}.

Proof. Recall that L ≅ SL_{2}(q) and V/C_{V} (L) is a natural SL_{2}(q)-module for L. Let A_{0}, …, A_{q} be the q + 1 Sylow p-subgroups of L. We set

q_{0} := |\overline{C_{V} (L)}|, L_{i} := ⟨A_{i}, B⟩, and V_{i} := C_{V} (L_{i}), for i = 0, …, q.

At least one of the groups L_{i} is non-abelian, so 4.8 implies that A_{i} is conjugate to B in L^{∗}. In particular |B| = q and B ∉ D_{P} (M). From 4.1 we get that |V/V_{i}| ≤ q^{2} and |\overline{V} | = q^{2}q_{0} ≤ q^{3}, so

|\overline{V}_{i}| ≥ q_{0} and |\overline{C_{V} (A_{i})}| = |\overline{C_{V} (B)}| = qq_{0}.

Suppose that (a) holds for some A ∈ D ∩ L. Then as L is generated by any two of its Sylow p-subgroups, A must be the unique element of D ∩ L which commutes with B. Furthermore, by 4.6 we get [V, A][V, B] ≤ C_{V} (AB) and |V/C_{V} (AB)| = q^{2}. Since V/C_{V} (L) is a natural SL_{2}(q)-module for L, this forces q_{0} = q = |[\overline{V} , A]| = |[\overline{V} , B]| and (b) - (e) hold.

It suffices to prove that (a) holds for some A ∈ D ∩ L, so we assume that [A, B] ≠ 1 for all A ∈ D ∩ L and aim for a contradiction.

Since V/C_{V} (L) is a natural SL_{2}(q) module for L and |V/C_{V} (A_{i})| = q, the subgroups C_{V} (A_{i})/C_{V} (L), 0 ≤ i ≤ q, form a partition of V/C_{V} (L). Thus

(∗)  V = ⋃_{i=0}^{q} C_{V} (A_{i}).

Hence for each b ∈ B^{♯} there exists a j ∈ {0, …, q} with [V, b] ∩ C_{V} (A_{j}) ≠ 1. Note that B and so also L_{j} centralizes [V, b] ∩ C_{V} (A_{j}). As B and A_{j} are conjugate in L_{j} we get [V, b] ∩ C_{V} (A_{j}) ≤ [V, b] ∩ [V, A_{j}]. Thus, we have:

4.13.1 For every b ∈ B^{♯}, there exists j ∈ {0, …, q} such that [V, b] ∩ [V, A_{j}] ≠ 1.

It follows from 4.6 that 1 ≠ [V, b] ∩ [V, D ∩ T^{g}] ≤ C_{V} (D ∩ T^{g}), where A_{j} ≤ T^{g}. Assume that there exists M_{j} ∈ M(A_{j}) \ M(B). By 4.9 H := C_{P} (C_{V} (D ∩ T^{g}) ∩ [V, b]) satisfies Hypothesis 4.2 with respect to H ∩M_{j}. But then by 4.12 there exists L̂ ∈ Λ(B) with L̂ ≤ H. By considering the action of L̂ on the natural SL_{2}(q)-module V/C_{V} (L̂) we get [V, b] ∩ C_{V} (L̂) = 1, which contradicts [V, b] ∩ C_{V} (D ∩ T^{g}) ≤ C_{V} (H). We have shown that M(A_{j}) ⊆ M(B), so M(B) = M(A_{j}), since A_{j} and B are conjugate. Recall that M(A_{j}) ≠ M(D) for every A_{j} ≠ D ∈ D ∩ L. Hence

4.13.2 C_{V} (A_{i}) ∩ [V, b] = 1 for every i ≠ j and b ∈ B^{♯}.

On the other hand, by 4.1 |[V, b]| = q. As the subgroups C_{V} (X)/C_{V} (L), X ∈ D ∩ L, form a partition of V/C_{V} (L), (∗) implies that [V, b] ≤ C_{V} (A_{j}) for every b ∈ B^{♯}. Using the Three Subgroups Lemma and the faithful action of P on V this gives [A_{j}, B] = 1, which is a contradiction. □

Theorem 4.14 Assume Hypothesis 4.2 and |D∩T | = 1. Then ⟨D⟩ ≅ SL_{2}(q), q = |A|, and V/C_{V} (⟨D⟩) is a natural SL_{2}(q)-module.

Proof. By 4.12 there exists L ∈ Λ and by 4.13 L = ⟨D⟩.

□

Lemma 4.15 Assume Hypothesis 4.3. Let A, B ∈ D ∩ T and L ∈ Λ(A) with L ≰ M and A ≠ B. Then [L, B] = 1.

Proof. Assume that [L, B] ≠ 1 and recall that [A, B] = 1 by 4.7. We apply 4.13 and use the notation given there. Then

(∗)  C_{V} (L^{∗})[V, A] = C_{V} (AB), |\overline{V} | = q^{3}, and |V/C_{V} (AB)| = q^{2}.

Let W := [V, L]C_{V} (L^{∗}). By 4.13 \overline{W} is a natural SL_{2}(q)-module for L and L^{∗}-invariant. For every 1 ≠ x ∈ AB and A ≠ D ∈ D ∩ L we have [L, D^{x}] ≠ 1, since [A, D^{x}] ≠ 1. Hence 4.13 also applies to L̂ := ⟨L, D^{x}⟩, if D^{x} ≰ L. In particular we get C_{V} (L^{∗}) = C_{V} (L̂) and [\overline{V} , D^{x}] = [\overline{V} , Y] for some Y ∈ D ∩ L with A ≠ Y. This shows that AB acts on the set

Ω_{0} := \{[\overline{V} , D] | D ∈ D ∩ L and D ≠ A\}.

As |Ω_{0}| = q and |AB| = q^{2}, we get that |N_{AB}([\overline{V} , D])| = q for D ∈ D ∩ L with A ≠ D. On the other hand, [\overline{V} , AB] = [\overline{V} , A], so C := C_{AB}(\overline{W}) has order q. Since [\overline{V} , L^{∗}] = \overline{W}, we conclude that C ≤ O_{p}(L^{∗}).

Let AB ≤ T_{0} ∈ Syl_{p}(L^{∗}). From 4.13 we get that C_{V} (AB) is T_{0}-invariant. Observe that C_{T_{0}}(C_{V} (AB)) ∩ C_{T_{0}}(V/C_{V} (AB)) is elementary abelian.

Hence 4.3 and 4.6 show that

AB = C_{T_{0}}(C_{V} (AB)) ∩ C_{T_{0}}(V/C_{V} (AB)),

so AB is normal in T_{0}. In particular [V, AB] = [V, A][V, B] is T_{0}-invariant. This gives

[V, D ∩ T, T_{0}] ≤ [C_{V} (AB), T_{0}] = [V, A, T_{0}] ≤ [V, AB] ≤ [V, D ∩ T],

so T_{0} normalizes [V, D ∩ T]. In particular, ⟨(D ∩ T)^{T_{0}}⟩ acts quadratically on V and so is p-group. Hence, T_{0} normalizes D ∩ T and T_{0} ≤ M. Then there exists x ∈ M with C_{V} (T^{x}) ≤ C_{V} (T_{0}), and [C_{V} (T_{0}), L] ≠ 1 since L ≰ M. This shows that W ≤ ⟨C_{V} (T_{0})^{L^{∗}}⟩, so O_{p}(L^{∗}) centralizes W and acts quadratically on V. In particular O_{p}(L^{∗}) is elementary abelian. Hence 4.3 implies C = O_{p}(L^{∗}) and thus [L, C] = 1. Now C_{V} (L) is AB-invariant and so C_{V} (L) ≤ C_{V} (AB). But then |V/C_{V} (AB)| = q which contradicts 4.1. □

Lemma 4.16 Assume Hypothesis 4.3. Let A, B ∈ D with [A, B] = 1 and A ∉ D_{P} (M). Then D ∩ AB = {A, B}.

Proof. We apply 4.12. Then, possibly after replacing A by a conjugate, we may assume that A ≤ T and that there exists L ∈ Λ(A) with L ≰ M. Hence, by 4.15 |D ∩ AC| = 2 for every C ∈ D ∩ T with C ≠ A. On the other hand, A, B ∈ D ∩ T^{g} for some g ∈ P, and by 4.7 D ∩ T^{g} and D ∩ T are both in C_{P} (A). Hence conjugation in C_{P} (A) gives the claim for |D ∩ AB|. □

Lemma 4.17 Assume Hypothesis 4.3. Let B ∈ D and L ∈ Λ with |X| ≥ 3 for every X ∈ D ∩ L. Then either B ≤ L or [L, B] = 1.

Proof. We may assume that [L, B] ≠ 1 and B ≰ L. As before we set

L^{∗} := ⟨L, B⟩ and \overline{V} := V/C_{V} (L^{∗}).

By 4.13 there exists a unique A ∈ D ∩ L such that

(∗)  [A, B] = 1 and C_{V} (AB) = C_{V} (L^{∗})[V, A].

We now use the fact that q := |A| ≥ 3. Let K be a complement for A in N_{L}(A). Then |K| = q − 1 ≥ 2 and by (∗) C_{V} (AB) is K-invariant. Hence A⟨B^{K}⟩ acts quadratically on V, and thus is abelian. On the other hand, by 4.1

|C_{V} (A)||A| = |V | = |C_{V} (AB)||AB| ≤ |C_{V} (AB)||A⟨B^{K}⟩|,

so 4.3 implies that AB = A⟨B^{K}⟩. In particular AB is K-invariant and by 4.16 K normalizes B and C_{V} (B).

Observe that K acts fixed-point-freely on the natural SL_{2}(q)-module V/C_{V} (L). Thus

\overline{V} = [\overline{V} , K] × \overline{C_{V} (K)} and C\overline{V} (K) = \overline{C_{V} (K)} = \overline{C_{V} (L)}.

It follows that \overline{C_{V} (K)} ∩ \overline{C_{V} (B)} = 1 and \overline{C_{V} (B)} ≤ [\overline{V} , K]. As \overline{C_{V} (B)} ∩ \overline{C_{V} (L)} ≤ \overline{C_{V} (L^{∗})} = 1, the action of K on V/C_{V} (L) yields either

\overline{C_{V} (B)} = [\overline{V} , A] or \overline{C_{V} (B)} = [\overline{V} , K].

In the first case C_{V} (B) = C_{V} (AB), which contradicts 4.1.

Thus we have \overline{C_{V} (B)} = [\overline{V} , K]. By 4.13 [\overline{V} , K] is L-invariant. It follows that ⟨B^{L}⟩ acts quadratically on V, so ⟨B^{L}⟩ is abelian. Now 4.1 (iii) shows that B is normal in L^{∗}, so [L, B] = 1, which contradicts our assumption. □

Theorem 4.18 Assume Hypothesis 4.3. Then there exist subgroups E_{1}, …, E_{r} of P such that the following hold for W_{i} := [V, E_{i}] and i, j ∈ {1, …, r} :

(a) D = D_{P} (M) ∪ (D ∩ E_{1}) ∪ ⋯ ∪ (D ∩ E_{r}) and ⟨D⟩ = ⟨D_{P} (M)⟩ × E_{1} × ⋯ × E_{r}.

(b) [W_{i}, E_{j}] = [W_{i}, ⟨D_{P} (M)⟩] = 1 for i ≠ j and V = W_{i}C_{V} (E_{i}).

(c) E_{i} ≅ SL_{2}(q_{i}) where q_{i} = |A| for A ∈ D ∩E_{i}, or E_{i} ≅ S_{m}, m odd, E_{i} ∩M ≅ S_{m−1}, and |A| = 2 for A ∈ D ∩ E_{i}.

(d) E_{i} ≅ SL_{2}(q_{i}) and W_{i}/C_{W_{i}}(E_{i}) is a natural SL_{2}(q_{i})-module for E_{i},or E_{i} ≅ S_{m} and W_{i} is a natural S_{m}-module for E_{i}. Moreover, in the second case D ∩ E_{i} acts as the conjugacy class of transpositions on W_{i}.

Proof. We will prove 4.18 by induction on |D| + |P |. Let P be a minimal counterexample. Then by 4.9: 4.18.1 P = ⟨D⟩. According to 4.8 and 4.10 there exists a partition of D satisfying 4.18.2 D = D_{0} ∪ D_{1} ∪ ⋯ ∪ D_{r} such that for E_{i} := ⟨D_{i}⟩ :

(1) D_{0} = D_{P} (M) and D_{i} ∩ D_{j} = ∅ for i ≠ j.

(2) [E_{i}, E_{j}] = 1 for i ≠ j, and D_{i} is a conjugacy class of E_{i} for i ≥ 1.

(3) For i ≥ 1, D_{i} and E_{i} satisfy Hypothesis 4.3 with respect to M ∩ E_{i}. Assume that D ≠ D_{i} for i ≥ 1. Then induction and 4.18.2 (3) show that (a) – (d) hold for E_{i} and D_{i}; in particular W_{i}/C_{W_{i}}(E) is an irreducible E_{i}-module. Hence [W_{i}, E_{j}] = 1 for i ≠ j, and (b) – (d) hold for P. Since W_{i}C_{V} (E_{i}) = V, we also get that E_{1} ⋯ E_{r} is the direct product of the subgroups E_{j} and also (a) holds. But then P is not a counterexample. We have shown: 4.18.3 D = D_{1} and P = E_{1}. Assume next that |A| ≥ 3 for A ∈ D. Then by 4.12 and 4.17 P ≅ SL_{2}(q) where q = |A|, and again (a) – (d) follow. Thus we have: 4.18.4 |A| = 2 = |V/C_{V} (A)| for A ∈ D. Then 4.7 and an elementary argument using dihedral groups yields 4.18.5 Let A ∈ D and D ∈ D \ C_{D}(A). Then L := ⟨A, D⟩ ≅ SL_{2}(2), and V/C_{V} (L) is a natural SL_{2}(2)-module for L.

Let A ∈ D ∩ T. According to 4.7 and 4.9 either C_{D}(A) ⊆ D ∩ M or C_{P} (A) satisfies Hypothesis 4.3 with respect to C_{D}(A) and C_{M}(A). In the first case by 4.18.5 there exists L ∈ Λ(A) with L ≰ M. Hence by 4.15 [B, L] = 1 for every B ∈ C_{D}(A) \ {A}, so

C_{D}(L) = C_{D}(D) \ {D} for every D ∈ D ∩ L.

Now 4.13 implies that P = L and P is not a counterexample. We have shown that

4.18.6 C_{D}(A) ⊈ D ∩ M; in particular C_{D}(A) ≠ {A} and C_{P} (A) satisfies Hypothesis 4.3 with respect to C_{D}(A) and C_{M}(A).

Let D_{A} := C_{D}(A) \ {A}. Assume first that D_{A} is not a conjugacy class of ⟨D_{A}⟩. Choose D^{∗} ⊆ D_{A} such that D^{∗} is a conjugacy class of ⟨D^{∗}⟩ and |D^{∗}| is maximal with that property. By our assumption there exists B ∈ D_{A} ∩ T with B ∉ D^{∗}, and by 4.8 [⟨D^{∗}⟩, B] = 1 for every such B. Hence the maximality of D^{∗} shows that ⟨D^{∗}⟩ is normal in ⟨D_{A}⟩ and ⟨D_{B}⟩.

Let D ∈ D with D ≰ M. Then M(D) ≠ M(B) and by 4.18.5 either D ∈ D_{B} or ⟨D, B⟩ ∈ Λ. In the former case D normalizes D^{∗} and in the latter case 4.15 implies that D ∈ D_{A}, so again D normalizes D^{∗}. It follows that

D = (D ∩ M) ∪ (D ∩ N_{P} (D^{∗})).

But then 2.1 shows that P = M or P = N_{P} (D^{∗}). The first case contradicts 4.2 and the second case contradicts D ≠ D^{∗} and the fact that D is a conjugacy class by 4.18.2 and 4.18.3. We have shown that D_{A} is a conjugacy class, so 4.18.4, 4.18.6 and induction give

4.18.7 ⟨D_{A}⟩ ≅ S_{n}, with n odd, M ∩ ⟨D_{A}⟩ ≅ S_{n−1}, W := [V, ⟨D_{A}⟩] is a natural S_{n}-module for ⟨D_{A}⟩, and D_{A} acts as the conjugacy class of transpositions on W.

Using the usual generators and relations for S_{n} we get from 4.18.7

4.18.8 There exist T_{1}, …, T_{n−1} ∈ D_{A} such that T_{i} ∈ D ∩ M for 1 ≤ i ≤ n − 2, and

[T_{i}, T_{j}] = 1 ⟺ |i − j| ≠ 1 and ⟨T_{i}, T_{j}⟩ ≅ SL_{2}(2) ⟺ |i − j| = 1.

By the same elementary observation as above D ⊈ M ∪ C_{P} (A). Hence by 4.18.5 there exists D ∈ D such that D ∉ M and ⟨A, D⟩ ∈ Λ(A). Now 4.15 gives

D ∩ M ∩ ⟨D_{A}⟩ ⊆ C_{D}(D);

in particular [D, T_{i}] = 1 for 1 ≤ i ≤ n − 2.

Set T_{n+1} := A and T_{n} := D. Then T_{1}, …, T_{n+1} generate a subgroup isomorphic to S_{n+2} provided we can show that [D, T_{n−1}] ≠ 1. Assume that [D, T_{n−1}] = 1. Then D_{A} = D_{D}, and as above 4.13, applied to ⟨A, D⟩, gives P = ⟨A, D⟩, and P is not a counterexample.

We have shown that T_{1}, …, T_{n+1} generate a subgroup U isomorphic to S_{n+2} in P. In particular C_{D}(X) ⊆ D ∩ U for every X ∈ D ∩ ⟨A, D⟩. Now 4.13 implies that P = U, and P is not a counterexample.

□

5 The Proof of the Local C^{∗∗}(G,T)-Theorem for Minimal Parabolic Groups

In this section we work with the following two hypotheses.

Hypothesis 5.1 Let p be a prime, P a minimal parabolic group acting faithfully on an elementary abelian p-group V, and let T ∈ Syl_{p}(P) and M ≤ P be the unique maximal subgroup of P containing T. Suppose also that:

(i) O_{p}(P) = 1,

(ii) O_{P} (V) ≠ ∅,^{2} and

(iii) C_{P} (C_{V} (T)) ≤ M (so [C_{V} (T), P] ≠ 1).

Hypothesis 5.2 Let P be a minimal parabolic group of characteristic p with T ∈ Syl_{p}(P) and C^{∗∗}(P, T) ≠ P, and let M be the unique maximal subgroup of P containing T.

Lemma 5.3 Assume Hypothesis 5.1. Then Hypothesis 4.3 holds for A_{P} (V); in particular |A||C_{V} (A)| = |V | for every A ∈ A_{P} (V). Moreover, N_{P} (A) acts irreducibly on V/C_{V} (A) for every A ∈ A_{P} (V).

Proof. From 3.6 we get that P satisfies Hypothesis 4.1 with respect to A_{P} (V) and that N_{P} (A) acts irreducibly on V/C_{V} (A) for every A ∈ A_{P} (V). In addition, since P is minimal parabolic and O_{p}(P) = 1, we also get Hypothesis 4.2. Now Hypothesis 4.3 follows from the definition of A_{P} (V). □

Lemma 5.4 Assume Hypothesis 5.2 and let

V := Ω(Z(O_{p}(P))) and \overline{P} := P/C_{P} (V).

Then \overline{P} and V satisfy Hypothesis 5.1, and

|A/C_{A}(V)||C_{V} (A)| = |V | for every A ∈ A(T)with A ≰ C_{P} (V).

2Here O_{P} (V) is the set introduced in 2.7.

Proof. Since C_{P} (O_{p}(P)) ≤ O_{p}(P) ≤ T, we have Ω(Z(T)) = C_{V} (T). Hence

C_{P} (V) ≤ C_{P} (C_{V} (T)) ≤ C^{∗∗}(P, T) ≤ M.

By Lemma 3.4 it follows that O_{p}(\overline{P}) = 1. It remains to show that O\overline{P} (V) ≠ ∅.

We first show that J(T) ≰ C_{P} (V). Suppose on the contrary that J(T) ≤ C_{P} (V). Then J(T) ≤ C_{T} (V) ∈ Syl_{p}(C_{P} (V)) and, as J(T) = J(C_{T} (V)) char C_{T} (V), the Frattini Argument gives

P = C_{P} (V)N_{P} (C_{T} (V)) ≤ C_{P} (Ω(Z(T)))N_{P} (J(T)) ≤ C^{∗∗}(P, T),

whichis a contradiction.

Therefore J(T) ≰ C_{P} (V) and there exists A ∈ A(T) with A ≰ C_{P} (V). Let A_{0} ≤ A. Then

|A| ≥ |A_{0}C_{V} (A_{0})| = \frac{|A_{0}||C_{V} (A_{0})|}{|A_{0} ∩ V |} ≥ \frac{|A_{0}||C_{V} (A_{0})|}{|C_{V} (A)|}.

Thus A ∈ O_{P} (V) and it follows immediately that \overline{A} ∈ O\overline{P} (V). Now 5.3 gives the additional statement. □

Theorem 5.5 Assume Hypothesis 5.1 holds. Let D := A_{P} (V). Then there exist subgroups E_{1}, …, E_{r} of P so that, for each 1 ≤ i ≤ r:

(a) P = (E_{1} × … × E_{r})T,

(b) T acts transitively on {E_{1}, …, E_{r}},

(c) D = (D ∩ E_{1}) ∪ ⋯ ∪ (D ∩ E_{r}),

(d) V = C_{V} (E_{1} × … × E_{r}) ∏_{i=1}^{r}[V, E_{i}], with [V, E_{i}, E_{j}] = 1,

(e) E_{i} ≅ SL_{2}(p^{n}) or p = 2 and E_{i} ≅ S_{2^{n}+1}, for some n ∈ ℕ, and

(f) [V, E_{i}]/C_{[V,E_{i}]}(E_{i}) is a natural module for E_{i}.

Proof. By 5.3, D satisfies Hypothesis 4.3, so we are allowed to apply 4.18 with the notation given there. Since P is minimal parabolic we get from 3.3 that O^{p}(P) ≤ E_{1} × … × E_{r} and as O_{p}(P) = 1, D_{P} (M) = ∅. Therefore (a) – (d) and (f) hold.

For the proof of (e) it suffices to show that m = 2^{n} + 1 if E_{i} ≅ S_{m}. Observe that N_{T} (E_{i})E_{i} = C_{T} (E_{i})E_{i}, so N_{T} (E_{i})E_{i} is a minimal parabolic group. Now (e) follows from 2.12 (e). □

The proof of the Local C^{∗∗}(G, T)-Theorem for minimal parabolic groups: Let \overline{P} := P/C_{P} (V). By 5.4 \overline{P} satisfies the hypothesis of 5.5. Thus the only thing that remains to be proven is

\overline{J(P)} = \overline{E}_{1} × ⋯ × \overline{E}_{r} =: \overline{E}.

Let A ∈ A(T). Suppose that \overline{A} ≰ \overline{E} and that |\overline{A}| is minimal with this property. By 5.3 and 5.4 there exists \overline{B} ≤ \overline{A} with \overline{B} ∈ A\overline{P} (V) and

|V | = |\overline{A}||C_{V} (\overline{A})| = |\overline{B}||C_{V} (\overline{B})|.

Moreover, N\overline{P} (\overline{B}) acts irreducibly on V/C_{V} (\overline{B}). The latter fact shows that there exists a unique k ∈ \{1, …, r\} such that \overline{B} ≤ \overline{E}_{k}.

Assume that \overline{E}_{k} ≅ SL_{2}(q). Then \overline{B} ∈ Syl_{p}(\overline{E}_{k}) and the structure of Aut(SL_{2}(q)) gives

\overline{A} = \overline{B} × \overline{A}_{0}, A_{0} := C_{A}(\overline{E}_{k}).

This shows that also A_{0}C_{V} (A_{0}) ∈ A(T), and the minimal choice of \overline{A} gives \overline{A}_{0} ≤ \overline{E}. But then also \overline{A} ≤ \overline{E}, which contradicts the choice of A.

Assume next that \overline{E}_{k} ≅ S_{2^{n}+1}. Then |\overline{B}| = 2 and by 2.16 (b)

\overline{A} = \overline{B} × \overline{A}_{0} with A_{0} ≤ A, and C_{V} (\overline{A}_{0}) ≰ C_{V} (\overline{B}).

Similarly, as in the previous case, this shows that A_{0}C_{V} (A_{0}) ∈ A(T) and then that \overline{A}_{0} ≤ \overline{E}.

Lemma 5.6 Let p be a prime and P be a minimal parabolic group acting faithfully on an elementary abelian p-group V. Suppose that O_{p}(P) = 1 and O_{P} (V) ≠ ∅. Then [C_{V} (T), P] ≠ 1 for every T ∈ Syl_{p}(P).

Proof. Let V_{0} := C_{V} (O^{p}(P)) and Ṽ := V/V_{0}. By 3.3 P also acts faithfully on Ṽ. We also have [CṼ(T), P] ≠ 1, for otherwise O^{p}(P) would centralize the inverse image of CṼ(T), contradicting the definition of V_{0}. Moreover, 2.8 shows that O_{P} (Ṽ) ≠ ∅. Hence (P, Ṽ) satisfies the hypothesis of 5.5, so we get (a) – (f) with Ṽ in place of V.

Let A ∈ O_{T} (V). Then there exists i ∈ {1, …, r} such that [E_{i}, A] ≠ 1. Hence 2.16 shows that A ≤ E_{i}C_{P} ([V, E_{i}])/C_{P} ([V, E_{i}]) and [V, E_{i}, A] ≤ C_{V} (T ∩ E_{i}); in particular C_{V} (T ∩ E_{i}) ≰ C_{V} (O^{p}(P)).

If E_{i} ≅ SL_{2}(p^{n}), with p^{n} > 2, then let K be a complement for T ∩ (E_{1} ⋯ E_{r}) in N_{E_{1}···E_{r}}(T ∩ (E_{1} ⋯ E_{r})). Then T = (T ∩ (E_{1} ⋯ E_{r}))N_{T} (K) and

C_{V} (O^{p}(P)) ∏_{i=1}^{r} C_{[V,E_{i}]}(T ∩ E_{i}) = C_{V} (O^{p}(P)) × [C_{V} (T ∩ (E_{1} ⋯ E_{r})), K].

Since N_{T} (K) normalizes [C_{V} (T ∩ (E_{1} ⋯ E_{r})), K], it follows that C_{V} (T) ≰ C_{V} (O^{p}(P)).

If E_{i} ≅ S_{2m+1}, then 2.13 shows that V = C_{V} (O^{2}(P)) × [V, O^{2}(P)] and again C_{V} (T) ≰ C_{V} (O^{p}(P)). □

6 B(T )-Blocks

In this section we assume

Hypothesis 6.1 Let G be of characteristic p and T ∈ Syl_{p}(G).

Notation 6.2 Let B(T) be the set of B(T)-blocks of G. We set

B(G) := ⋃_{g∈G} B(T^{g}).

Moreover, B^{∗}(G) is the set of maximal elements of B(G) with respect to inclusion and

B^{∗}(T) := B^{∗}(G) ∩ B(T).

For E ∈ B(G) we set W_{E} := [Ω(Z(O_{p}(E))), E].

Lemma 6.3 Let E ∈ B(T). Suppose that Q is a p-subgroup of G normalized by B(T)E. Then Q ≤ N_{G}(E).

Proof. As B(B(T)Q) = B(T), Q normalizes B(T). Moreover, from E = [E, B(T)] we get that EB(T) = ⟨B(T)^{E}⟩. Hence Q normalizes EB(T) and thus also E = O^{p}(EB(T)).

□

Lemma 6.4 Let E ∈ B(T). Then the following hold:

(a) E = O^{p}(EO_{p}(G)) and W_{E} ≤ Ω(Z(O_{p}(G))).

(b) Assume that E is not exceptional. Then

O_{p}(E) ≤ Ω(Z(O_{p}(G))) and [O_{p}(G), E] = W_{E}.

(c) Assume that E is exceptional. Then Z(E)W_{E} = Ω(Z(O_{3}(E))) ≤ Ω(Z(O_{3}(G)))

and either

O_{3}(E) ≤ O_{3}(G) or [O_{3}(G), E] = W_{E}.

(d) [W_{E}, J(T)] ≠ 1.

Proof. (a): From 6.3 with Q := O_{p}(G) we get O_{p}(G) ≤ N_{G}(E). The first part now follows from the fact that E = O^{p}(E). Since W_{E}Z(E)/Z(E) is an irreducible E-module, [W_{E}, O_{p}(G)] ≤ Z(E). Hence the Three Subgroups Lemma gives

[W_{E}, O_{p}(G)] = [W_{E}, E, O_{p}(G)] = 1,

so W_{E} ≤ Ω(Z(O_{p}(G))), since G is of characteristic p.

(b): Note that W_{E} = O_{p}(E) and W_{E} = [W_{E}, E], so the result follows from (a). (c): Since [O_{3}(E), O_{3}(G)] ≤ Ω(Z(O_{3}(E))) the Three Subgroups Lemma gives

[O_{3}(E), O_{3}(E), O_{3}(G)] = 1.

It follows that Z(E) ≤ Ω(Z(O_{3}(G))) and by (a)

Z(E)W_{E} = Ω(Z(O_{3}(E))) ≤ Ω(Z(O_{3}(G))).

The other statement in (c) is a direct consequence of the structure of O_{3}(E) and the fact that E = O^{3}(E).

(d): From the definition of a B(T)-block we get E = [E, B(T)] and [W_{E}, E] ≠ 1. Hence W_{E} ≰ Z(B(T)) and (d) follows. □

Lemma 6.5 Let E ∈ B(T) be an exceptional B(T)-block. Then

O^{3}(N_{G}(E) ∩ C_{G}(W_{E})) ≤ C_{G}(E).

Proof. We fix the following notation:

\begin{aligned}R & := O^{3}(N_{G}(E) ∩ C_{G}(W_{E})), & M & := N_{G}(E) ∩ C_{G}(Z(E)), \\ M_{2} & := C_{M}(O_{3}(E)/Z(E)W_{E}), & \overline{N_{G}(E)} & := N_{G}(E)/C_{G}(E). \end{aligned}

We first show:

(∗)  \overline{M}/O_{3}(\overline{M}) ≅ SL_{2}(3^{n}).

We put E^{∗} := E if E is non-solvable. If E/O_{3}(E) ≅ Q_{8}, then there exists a ∈ B(T) such that E⟨a⟩/O_{3}(E) ≅ SL_{2}(3) and we put E^{∗} := E⟨a⟩. Then 2.6 applies to E^{∗} and we get \overline{M} = \overline{E}^{∗}\overline{M}_{2}.

Note that \overline{E} ∩ \overline{M}_{2} ≤ O_{3}(\overline{E}) ≤ O_{3}(\overline{M}). Moreover, C_{\overline{M}_{2}}(W_{E}) centralizes an E-chief series of E, so C_{\overline{M}_{2}}(W_{E}) ≤ O_{3}(\overline{M}). Hence Schur’s Lemma implies that \overline{M}_{2}/O_{3}(\overline{M}) is a cyclic group whose order divides 3^{n} − 1. In particular, M_{2} normalizes C_{W_{E}}(B(T)) and so [\overline{B(T)}, \overline{M}_{2}] ≤ O_{3}(\overline{M}). This shows that B(T)C_{G}(E) is normalized by M_{2}.

If \overline{M}_{2} = O_{3}(\overline{M}), then \overline{M} = \overline{E}^{∗}O_{3}(\overline{M}) and (∗) follows. So assume that \overline{M}_{2} ≠ O_{3}(\overline{M}). Then there exists a non-trivial 3′-subgroup \overline{Q} ≤ \overline{M}_{2} and this subgroup normalizes \overline{B(T)}. Hence

\overline{B(T)} = \overline{A}(\overline{B(T)} ∩ O_{3}(\overline{M})), with \overline{A} := C\overline{B(T)}(\overline{Q}).

But then A leaves invariant the decomposition

O_{3}(\overline{E}) = C_{O_{3}(\overline{E})}(\overline{Q}) × \overline{W}_{E},

and acts quadratically in each factor. This contradicts the definition of an exceptional B(T)-component and finishes the proof of (∗).

According to (∗), \overline{R}∩\overline{M} ≤ O_{3}(\overline{M}). Thus we may assume that \overline{R} ≰ \overline{M}, for otherwise the result follows. Consider R_{0} := C_{R}(O_{3}(E)/W_{E}Z(E)). Then [O_{3}(E), R_{0}] ≤ Z(E)W_{E} and the Three Subgroups Lemma yields

[O_{3}(E), O_{3}(E), R_{0}] = [Z(E), R_{0}] = 1,

so \overline{R}_{0} ≤ O_{3}(\overline{M}). Again Schur’s Lemma shows that \overline{R}/\overline{R} ∩ O_{3}(\overline{M}) is a cyclic 3′-group. Let \overline{Q} be a non-trivial 3′-subgroup of \overline{R}.

As B(T) normalizes R, we get

[\overline{R}, \overline{B(T)}] ≤ \overline{R} ∩ \overline{M} = O_{3}(\overline{M}).

It follows that \overline{R} normalizes \overline{B(T)}. In particular

\overline{B(T)} = \overline{A}(\overline{B(T)} ∩ O_{3}(\overline{M})) with \overline{A} := C\overline{B(T)}(\overline{Q}).

As in the proof of (∗), this contradicts the definition of an exceptional B(T)-block. □

Theorem 6.6 Let E ∈ B(T). Then E ⊴ EC_{G}(W_{E}).

Proof. We fix the following notation:

W := W_{E}, C := C_{G}(W), C_{0} := C_{G}(O_{3}(E)/Z(E)W), R := [C, E], \overline{G} := G/O_{3}(G).

Let G be a minimal counterexample. Then G = CEB(T) and W ⊴ G. We will prove the result in a sequence of steps.

6.6.1 E is exceptional and O_{3}(E) ≤ O_{3}(G); in particular E/O_{3}(E) ≅ SL_{2}(q)′, q = 3^{n}.

Assume that E is not exceptional or p = 3 and O_{3}(E) ≰ O_{3}(G). Then by 6.4 [O_{p}(G), E] ≤ W. Hence [E, C_{G}(W)] centralizes W and O_{p}(G)/W, so

[E, C_{G}(W)] ≤ O_{p}(G).

Now 6.4 (a) implies that E is normal in EC_{G}(W) and G is not a counterexample.

We now fix in addition an involution t ∈ E with [t, E] ≤ O_{3}(E) and O_{3}(G) ≤ Y ≤ C such that \overline{Y} = C\overline{C}(\overline{t}). Note that Y = C_{Y} (t)O_{3}(G).

6.6.2 Let N ≤ C be an EB(T)-invariant subgroup. Then either C = N(C ∩ EB(T)) and O^{3}(C) ≤ N, or N ≤ N_{G}(E).

If NEB(T) < G, then by induction N ≤ N_{G}(E), and if NEB(T) = G, then C = N(C ∩ EB(T)). Since C ∩ EB(T) ≤ O_{3}(EB(T)), the latter case gives O^{3}(C) ≤ N.

6.6.3 O^{3}(\overline{C}) = F^{∗}(\overline{C}), and O^{3}(C) ≰ N_{G}(E).

Let F be the inverse image of F^{∗}(\overline{C}) in G. Assume first that F ≤ N_{G}(E). Then by 6.5 O^{3}(F) ≤ C_{G}(E), so [\overline{F} , \overline{E}] = 1. It follows that \overline{R} ≤ C\overline{C}(\overline{F}) ≤ \overline{F}. Hence R ≤ F and O^{3}(R) ≤ C_{G}(E). Now 2.5 (with N := R) implies that E is normal in G, a contradiction.

We have shown that F ≰ N_{G}(E), and thus by 6.6.2 O^{3}(\overline{C}) = F^{∗}(\overline{C}).

6.6.4 Either C = Y, or O^{3}(\overline{C}) is an r-group, r a prime different from 2 and 3.

Note that Y is EB(T)-invariant. Hence by 6.6.2 either C = Y (C ∩ EB(T)) or Y ≤ N_{G}(E). As [t, EB(T)] ≤ O_{3}(E) ≤ O_{3}(G), the first case gives C = Y.

Assume that Y ≤ N_{G}(E). Then

[\overline{Y} , \overline{E}] ≤ \overline{Y} ∩ \overline{E} ≤ \overline{C} ∩ \overline{E} = 1,

since W is a faithful \overline{E}-module. It follows that \overline{Y} = C\overline{C}(\overline{S}), where \overline{S} is a Sylow 2subgroup of \overline{E}. As \overline{S} is a quaternion group we conclude from 2.4 that \overline{U} := [\overline{C}, \overline{t}] is solvable of odd order. In particular \overline{C} = \overline{Y} \overline{U}, so the inverse image U is not in N_{G}(E). As U is EB(T)-invariant, 6.6.2 yields C = U(C ∩ EB(T)), and thus O^{3}(C) ≤ U. Now 6.6.3 shows that \overline{U} = F (\overline{C}). Let r be a prime dividing |\overline{U}|, so r ∉ \{2, 3\}. Then, again using 6.6.2, \overline{U} = O_{r}(\overline{C}).

6.6.5 C ≠ Y, so O^{3}(\overline{C}) is an r-group, r a prime different from 2 and 3.

Assume that C = Y. Then C = C_{C}(t)O_{3}(G) and both O_{3}(G) and C_{C}(t) normalize [O_{3}(G), t] = O_{3}(E). From G = CEB(T) we conclude that O_{3}(E) ⊴ G. By 2.6, EC_{0} is normal in G, so R ≤ EC_{0}.

Note that R centralizes O_{3}(G)/O_{3}(E), Z(E) and W, so R ∩ C_{0} ≤ O_{3}(G). It follows that either [E, R] ≤ O_{3}(G) or t ∈ RC_{0}.

In the first case by 6.4 (a) R ≤ N_{G}(E), and thus by 6.5 O^{3}(R) ≤ C_{G}(E). Now 2.5 shows that G is not a counterexample.

In the second case there exists an involution a ∈ R such that t ∈ aC_{0} and [a, E] ≤ R ∩ C_{0} ≤ O_{3}(G). Now again 6.4 (a) and 6.5 give a ∈ C_{G}(E), and a centralizes O_{3}(G)/O_{3}(E) and O_{3}(E), which contradicts the fact that G is of characteristic 3.

We derive a final contradiction. Let Q := [O_{3}(E), G], D := Φ(Q), and Q̃ := Q/W D. Note that O_{3}(G) centralizes Q̃, so \overline{G} acts on Q̃. The action of t on Q shows that

Q̃ = [Q̃, E] × CQ̃(E) and [Q̃, E] = { \atop O_{3}(E)}.

If [Q̃, E] = 1, then O_{3}(E) ≤ W D, and thus Q = O_{3}(E) = W, which is impossible. Hence [Q̃, E] is a natural SL_{2}(3^{n})′-module for E.

Let A := T ∩ E and C_{1} be the inverse image of O^{3}(C) in G. Then \overline{A} acts quadratically on Q̃ and CQ̃(A) = CQ̃(\overline{a}) for every \overline{a} ∈ \overline{A}^{♯}. Recall from 6.6.5 that \overline{C}_{1} is a 3′-group.

Assume first that q > 3. Then \overline{C}_{1} = ⟨C_{\overline{C}_{1}}(\overline{a}) | \overline{a} ∈ \overline{A}^{♯}⟩ and each C_{\overline{C}_{1}}(\overline{a}) normalizes CQ̃(A) = CQ̃(\overline{a}). Hence [\overline{C}_{1}, A] = [\overline{C}_{1}, A, A] centralizes CQ̃(A) and Q̃/CQ̃(A). As [C, A] also centralizes W and O_{3}(G)/Q, we conclude that O^{3}([C_{1}, A]) centralizes O_{3}(G), and thus [C_{1}, A] ≤ O_{3}(G). But then also [C_{1}, E] ≤ O_{3}(G), which using 6.4(a) implies that C_{1} normalizes E. This contradicts 6.6.3.

Assume now that q = 3, so E/O_{3}(E) ≅ Q_{8}. For x ∈ C_{1} set L := ⟨E, E^{x}⟩. Then either [E, x] ≤ O_{3}(G), and thus x ∈ N_{G}(E), or C_{1} ∩ L ≰ O_{3}(G). According to 6.6.3 we may assume that C_{1} ∩ L ≰ O_{3}(G).

Observe that L acts on Q̃_{0} := [Q̃, t][Q̃, t^{x}] and |Q̃_{0}| ≤ 3^{4}. Let L_{0} be the kernel of this action. If L ∩ C_{1} ≰ L_{0}, then by the order of GL_{4}(3) and 6.6.5, L ∩ C_{1}/L_{0} ∩ C_{1} is a cyclic group of order 5 or 13 which is normalized by E/O_{p}(E) ≅ Q_{8}, but it is easily checked that this is impossible in GL_{4}(3). Therefore L ∩ C_{1} ≤ L_{0}. Hence O^{3}(L ∩ C_{1}) centralizes the L-series D ≤ DW ≤ Q_{0} ≤ O_{3}(G), and thus L ∩ C_{1} ≤ O_{3}(L). But this contradicts L ∩ C_{1} ≰ O_{3}(G) and 6.6.5. □

Lemma 6.7 Let E ∈ B(T) and F ∈ B(G) such that [E, F] ≤ E. Then either F = E, or [F, E] = 1, or p = 2 and the following hold:

(a) F ≤ E and O_{2}(F) ≤ O_{2}(E).

(b) F O_{3}(E)/O_{2}(E) ≅ A_{2r+1} and E/O_{2}(E) ≅ A_{2m+1}, for some r ≤ m.

(c) There exists g ∈ E such that E, F ∈ B(T^{g}).

Proof. If [W_{E}, F] = 1, then 6.5 implies [E, F] = 1 and if [W_{F} , E] = 1, then by 6.6 E ≤ N_{G}(F) and again 6.5 implies [E, F] = 1. Thus we may assume that [W_{E}, F] ≠ 1 and [W_{F} , E] ≠ 1. As W_{E} is normalized by F, we get that W_{F} ≤ W_{E}.

We fix the following notation:

R := C_{G}(W_{E})E, \overline{F R} := F R/C_{G}(W_{E}), R̃ := R/O_{p}(R).

Then \overline{F} induces automorphisms in \overline{E} ≅ SL_{2}(p^{n})′ or A_{2m+1}.

6.7.1 The case F ≤ R.

Let F_{0} := EF ∩ C_{G}(W_{E}). Then

ẼF̃ = Ẽ × F̃_{0}.

By 6.6 C_{G}(W_{E}) ≤ C_{G}(W_{F}) ≤ N_{G}(F), so [F, F_{0}] ≤ O_{p}(F) and F′ ≤ EO_{p}(F). It follows that F̃ ≤ Ẽ, or one of the following two cases holds:

(i) p = 2 and F/O_{2}(F) ≅ C_{3}, or

(ii) p = 3, F/O_{3}(F) ≅ Q_{8} and F E/O_{3}(F E) ≅ SL_{2}(3^{m})′ × C_{2} × C_{2}.

In case (i) neither E nor F are exceptional. Hence 6.4 (b), applied to E and F, gives [O_{2}(G), EF] = W_{E}. Then [O_{2}(G), O^{2}(F_{0}), O^{2}(F_{0})] = 1. As G is of characteristic 2, this shows that F_{0} is a 2-group and F̃ ≤ Ẽ.

In case (ii) let t be an involution in F. Then t ∈ O^{3}(F′) ≤ E and [t, E] ≤ O_{3}(E). It follows that O_{3}(E) = [O_{3}(E), t] = O_{3}(F) and [O_{3}(G), EF] ≤ O_{3}(E). If E is exceptional, then 6.5 implies [E, O^{3}(F_{0})] = 1, so [O_{3}(G), O^{3}(F_{0}), O^{3}(F_{0})] = 1. If E is not exceptional, then O_{3}(E) = W_{E} and again [O_{3}(G), O^{3}(F_{0}), O^{3}(F_{0})] = 1. Thus, we have the same property as in case (i). As there we get that F̃ ≤ Ẽ.

Thus, in all cases we have established that F̃ ≤ Ẽ. Now 6.4 (a) implies E = O^{p}(EO_{p}(R)), and thus F = O^{p}(F) ≤ E.

Suppose that E is a linear block. Then the p′-elements of \overline{E} act fixed-point-freely on W_{E}/C_{W_{E}}(E). It follows that W_{E} = W_{F} C_{W_{E}}(E), and thus E = F.

Suppose E is a symmetric block. We first treat the case where F is a linear block, so \overline{F} ≅ SL_{2}(2^{k})′. Suppose k > 1. Then by 2.14 (c) k = 2 and there exists g ∈ E such that J(T)^{g} normalizes W_{F} and C_{W_{F}} (F). Put W̌_{F} = W_{F} /C_{W_{F}} (E). By 2.16 (b) there exist elements in J(T)^{g} acting as transvections on W̌_{F}. On the other hand, F ∈ B(T^{h}) for some h ∈ G. So J(T^{h}) normalizes F and W̌_{F}. It follows that J(T^{h}) acts GF (4)- semilinearly on W̌_{F} and so no element of J(T^{h}) acts as a transvection on W̌_{F}. But J(T^{h}) and J(T^{g}) are conjugate in N_{G}(W̌_{F}), a contradiction. This contradiction gives k = 1, so F is also a symmetric block.

We have shown that F is always a symmetric block; in particular (a) and (b) hold. By 2.16 (b), (e) \overline{B(T)} is generated by a maximal set of commuting transpositions on W_{E}. Hence 2.14 (b) implies (c).

6.7.2 The case F ≰ R.

Since both C_{Aut(W_{E})}(\overline{E}) and Out(\overline{E}) are solvable, \overline{F E}/\overline{E} is solvable. Thus \overline{F} ≰ \overline{E} implies F ≠ F′, so p = 2, 3 and F/O_{p}(F) ≅ SL_{2}(p)′. Moreover, if E is a symmetric block, then |\overline{F E}/\overline{E}| ≤ 2, while |F/O_{2}(F)| = 3, a contradiction. Hence \overline{E} ≅ SL_{2}(p^{k}) with k > 1, O_{p}(\overline{F}) ≤ \overline{E} and by 3.7 \overline{B(T)} ∈ Syl_{p}(\overline{E}). In particular W_{F} C_{W_{E}}(E) < W_{E}.

Assume that [W_{F} , B(T)^{h}] = 1 for some h ∈ E. As F ∈ B(G), there exists g ∈ G such that F ∈ B(T^{g}); so [W_{F} , B(T^{g})] ≠ 1 while [W_{F} , B(T)^{h}] = 1. But this is impossible since B(T)^{h} and B(T^{g}) are conjugate in N_{G}(W_{F}). We have shown that [W_{F} , B(T)^{h}] ≠ 1 for every h ∈ E.

If O_{p}(\overline{F}) ≠ 1, then \overline{F} normalizes a Sylow p-subgroup of \overline{E} and thus a conjugate \overline{B(T)}\overline{x}, x ∈ E. If O_{p}(F) ≤ C_{G}(W_{E}), then F/C_{F} (W_{E}) is a p′-group and C_{W_{E}}(F) ≰ C_{W_{E}}(E). Hence also in this case \overline{F} normalizes a Sylow p-subgroup of \overline{E} and thus a conjugate \overline{B(T)}\overline{x}, x ∈ E.

As we have seen above W_{F} ≰ C_{W_{E}}(B(T)^{h}) for every h ∈ E. Since W_{F} is an irreducible F -module, we get from the module structure of W_{E}

(∗)  [C_{W_{E}}(B(T)^{x}), F] = 1 and W_{E} = W_{F} × C_{W_{E}}(B(T)^{x}).

In particular \overline{E} ≅ SL_{2}(p^{2}) and O_{p}(\overline{F}) = 1. As |Syl_{p}(\overline{E})| = 5 resp. 10 and |\overline{F} | = 3 resp. 8, there exists a second conjugate \overline{B(T)}\overline{y}, y ∈ E, normalized by \overline{F}. But then also [C_{W_{E}}(B(T)^{y}), F] = 1, which contradicts (∗) since W_{E} = C_{W_{E}}(B(T)^{y})C_{W_{E}}(B(T)^{x}).

□

Lemma 6.8 Let E ∈ B(T) be a symmetric block with E ≰ C^{∗}(G, T). Then there exists F ∈ B(T) such that F ≤ E, F ≰ C^{∗}(G, T) and F/O_{2}(F) ≅ A_{3} ≅ SL_{2}(2)′.

Proof. Note that A ∈ A(T) satisfies (∗) of 2.16. Hence by 6.4 (d) and 2.16 (b) there exist A ≤ B(T) and E^{∗} = EA such that Ẽ^{∗} := E^{∗}/O_{2}(E^{∗}) ≅ S_{2n+1} and Ã is generated by a maximal set of commuting transpositions. We can choose d̃ ∈ Ẽ of order 3 to be inverted by one of these transpositions and commute with the others such that d ∉ C^{∗}(G, T). Then F := ⟨d⟩[W_{E}, d] has the required properties. □

Lemma 6.9 Let B(T)_{max} be the set of maximal elements of B(T). Then

B(T)_{max} = B^{∗}(T).

Proof. Let F ∈ B(T)_{max} and F ≤ E ∈ B^{∗}(G). By 6.7 (c) there exists g ∈ G such that F, E ≤ B(T^{g}). Then there exists h ∈ N_{G}(F) such that B(T^{gh}) = B(T). Hence F ≤ E^{h} ∈ B(T), so F = E^{h}, since F ∈ B(T)_{max}. It follows that E = F and F ∈ B^{∗}(G).

□

Lemma 6.10 Let E ∈ B^{∗}(T). Then E is the unique element of B^{∗}(G) in EC_{G}(W_{E}) that is not contained in C_{G}(W_{E}).

Proof. Let F ∈ B^{∗}(G) and F ≤ EC_{G}(W_{E}). Then by 6.6 [E, F] ≤ E, and thus by 6.7 either [E, F] = 1 or F ≤ E. In the latter case the maximality of F implies F = E.

□

Lemma 6.11 Let E, F ∈ B(G). Suppose that E and F are subnormal in G. Then E = F or [E, F] = 1.

Proof. Let \overline{G} := G/O_{p}(G). The subnormality of E implies that either \overline{E} is a component of \overline{G} or \overline{E} ≤ F (\overline{G}).

If [\overline{E}, \overline{F}] ≤ \overline{E}∩\overline{F}, then by 6.4 (a) [E, F] ≤ E∩F, and 6.7 gives E = F or [E, F] = 1. Thus we may assume

(∗)  [\overline{E}, \overline{F}] ≰ \overline{E} ∩ \overline{F} .

In particular, (∗) shows that E and F are both solvable, so \overline{E} ≅ \overline{F} ≅ C_{3} or Q_{8}.

Let L := ⟨E, F ⟩ and W := [Ω(Z(O_{p}(G)), L]. Then C_{L}(W) ≤ O_{2}(L), since C_{L}(W) centralizes O_{2}(G)/W and W. As L is also subnormal in G, we get \overline{C_{L}(W)} = 1.

Assume first that \overline{E} ≅ C_{3}. Then by 6.4 |W | ≤ 2^{4} and [E, F] ≤ C_{L}(W) since GL_{4}(2) has abelian Sylow 3-subgroups. Thus \overline{C_{L}(W)} = 1 gives [\overline{E}, \overline{F}] = 1, which contradicts (∗).

Assume that \overline{E} ≅ Q_{8}. If Z(\overline{E}) is normal in \overline{L}, then also [W, Z(\overline{E})] = W_{E} is Linvariant. As GL_{2}(3) \ SL_{2}(3) does not contain elements of order 4, F normalizes EC_{G}(W_{E}), and thus by 6.10 also E. But this contradicts (∗).

Suppose that Z(\overline{E}) is not normal in \overline{L}. There exists y ∈ L such that E^{y} ≠ E but [E, E^{y}] ≤ E ∩ E^{y}. Hence as already seen, [E, E^{y}] = 1 and \overline{E} × \overline{E}^{y} ≅ Q_{8} × Q_{8}. On the other hand, similarly to the above, \overline{L} is a subgroup of SL_{4}(3). Since a Sylow 2-subgroup of SL_{4}(3) has order 2^{8}, we get that \overline{F} ∩ (\overline{E} × \overline{E}^{y}) ≠ 1. Hence Z(\overline{F}) ≤ Z(\overline{E}) × Z(\overline{E}^{y}) and thus Z(\overline{F}) = Z(\overline{E}) or Z(\overline{E}^{y}). In both cases Z(\overline{E}) is normal in \overline{L}, a contradiction.

□ Theorem 6.12 Let E ∈ B(G). Suppose that E is subnormal in G. Then the following hold:

(a) E ⊴ B(G).

(b) E ∈ B(T^{x}) for every x ∈ G.

(c) For every F ∈ B(G) either F ≤ E or [F, E] = 1.

Proof. Observe that E ∈ B^{∗}(G), since E is subnormal in G. Let

V := ⟨Ω(Z(T))^{G}⟩, \overline{G} := G/C_{G}(V).

(a): We may assume that E is a B(T)-block. By 2.16 J(T^{x}) ≤ N_{G}(\overline{E}) for all x ∈ G, so by 6.10 J(T^{x}) also normalizes E. It follows that W_{E} ∩ Z(B(T^{x})) ≰ Z(E), so [W_{E}, E^{y}] ≠ 1 for all y ∈ B(T^{x}). Now 6.11 implies that B(T^{x}) ≤ N_{G}(E). Hence

E ⊴ ⟨B(T^{x}) | x ∈ G⟩ = B(G).

(b): For every x ∈ G, B(T) and B(T^{x}) are conjugate in B(G). Thus, (a) implies (b).

(c): Let F ∈ B(G). By (a) F normalizes E. Now 6.7 shows that F ≤ E or [E, F] = 1. □

7 The Proof of the Local C^{∗}(G,T)-Theorem

In this section we investigate a minimal counterexample to the Local C^{∗}(G, T)-Theorem. We assume in this section:

Hypothesis 7.1 Let G be a group of characteristic p with T ∈ Syl_{p}(G) such that G is a minimal counterexample to the Local C^{∗}(G, T)-Theorem.

Notation 7.2 We use the notation introduced in 6.2. In addition we define

\begin{aligned}B_{∗}(T) & := \{E ∈ B^{∗}(T) | E ≰ C^{∗}(G, T)\}, B_{∗}(G) := ∪_{g∈G}B_{∗}(T^{g}), \\ V & := ⟨Ω(Z(T))^{G}⟩, Z := Ω(Z(B(T))), \overline{G} := G/C_{G}(V). \end{aligned}

Observe that O_{p}(\overline{G}) = 1 (see for example [13, 2.0.1]).

Moreover, L(T) is the set of proper subgroups L < G satisfying:

B(T) ≤ L and L ≰ C^{∗}(G, T).

Set L(G) := ∪_{g∈G}L(T).

Lemma 7.3 Every L ∈ L(G) satisfies the hypothesis and conclusion of the Local C^{∗}(L, S)-Theorem for S ∈ Syl_{p}(L).

Proof. This follows from 2.3 and the minimality of G as a counterexample. □

Lemma 7.4 Let E ∈ B_{∗}(G). Then E is not subnormal in G.

Proof. Let Ω be the set of all elements in B_{∗}(G) that are subnormal in G and assume that Ω ≠ ∅. We will show that G is not a counterexample to the Local C^{∗}(G, T)- Theorem. Set

G_{0} := ∏_{E∈Ω} E, R := C_{G}([V, G_{0}]).

Clearly no element of Ω is contained in R; in particular RT is a proper subgroup of G. Now 7.3 implies that R ≤ C^{∗}(G, T), since R is normal in G.

By 6.12 G satisfies (a), (b), and (d) of the Local C^{∗}(G, T)-Theorem; in particular G_{0} ⊴ G. It remains to show (c) and (e) to get the desired contradiction.

Let E, Ẽ ∈ Ω with E ≠ Ẽ. Then by 6.3 [V, E] ≤ E and by 6.12 (c) Ẽ ≤ C_{G}([V, E]). The Dedekind identity then yields

EC_{G}([V, E]) ∩ ẼC_{G}([V, Ẽ]) = EẼC_{G}([V, E][V, Ẽ]).

Now an elementary induction argument shows that

⋂_{E∈Ω}(B(T)EC_{G}([V, E])) = B(T) ⋂_{E∈Ω}(EC_{G}([V, E])) = B(T)G_{0}R.

Let x ∈ G. By 6.12 and 2.16

B(T)^{x} ≤ B(T)EC_{G}([V, E]) for every E ∈ Ω.

It follows that B(T)G_{0}R = B(T)^{x}G_{0}R, and B(T)G_{0}R is normal in G. So the Frattini argument gives

G = G_{0}RN_{G}(B(T)) = G_{0}C^{∗}(G, T).

Thus also (c) of the Local C^{∗}(G, T)-Theorem holds.

Using 6.12 and 2.16 we get that B(T)E/O_{p}(B(T)E) ≅ SL_{2}(p^{m}) or S_{2m+1} for E ∈ Ω. In the first case B(T) ∈ Syl_{p}(B(T)E) and N_{B(T)E}(B(T)) is a maximal subgroup of B(T)E. In the second case N_{B(T)E}(B(T)) = N_{B(T)E}(Y), where Y O_{2}(E)/O_{2}(E) is a subgroup of S_{2m+1} generated by a maximal set of commuting transpositions. Furthermore, we get from 2.13 that W := [V, E]ΩZ(T) = C_{W} (E) × [V, E] and then from 2.11 that

⟨N_{B(T)E}(B(T)), C_{B(T)E}(Ω(Z(T ∩ B(T)E)))⟩/O_{2}(B(T)E) ≅ A_{2m};

in particular ⟨N_{B(T)E}(B(T)), C_{B(T)E}(Ω(Z(T ∩ B(T)E)))⟩ is a maximal subgroup of B(T)E. We conclude that in both cases C^{∗}(G, T) ∩ B(T)E is a maximal subgroup of B(T)E since B(T)E ≰ C^{∗}(G, T). Now also (e) of the Local C^{∗}(G, T)-Theorem holds. But then G is not a counterexample. □

Lemma 7.5 G is not a minimal parabolic group.

Proof. Assume that G is minimal parabolic. Then G satisfies the hypothesis of the C^{∗∗}(G, T)-Theorem for minimal parabolic groups because C^{∗∗}(G, T) ≤ C^{∗}(G, T). Hence, we can apply this theorem to G, as it was already proven in Chapter 5.

Let U := Ω(Z(O_{p}(G))). Then there exists a subnormal subgroup E_{1} of G with

E_{1} ≰ C^{∗}(G, T) and C_{G}(U) ≤ E_{1}

such that

E_{1}/C_{E_{1}}(U) ≅ SL_{2}(p^{n}) or S_{2^{m}+1} (and p = 2),

and [U, E_{1}]/C_{[U,E_{1}]}(E_{1}) is the corresponding natural module. Moreover, every other conjugate of E_{1} in G centralizes [U, E_{1}], and U = C_{U}(E_{1})[U, E_{1}]. As C_{[U,E_{1}]}(J(T)) ≰ C_{[U,E_{1}]}(E_{1}), this gives B(T) ≤ N_{G}(E_{1}).

Let H := B(T)E_{1} and W := Ω(Z(O_{p}(H))). Note that [O_{p}(H), E_{1}] ≤ O_{p}(E_{1}) ≤ O_{p}(G) and that [U, E_{1}] = [U, E_{1}, E_{1}] since U = C_{U}(E_{1})[U, E_{1}]. As [U, E_{1}]/C_{[U,E_{1}]}(E_{1}) is irreducible, the Three Subgroups Lemma yields that [U, E_{1}] ≤ W.

By 2.3 (c) H is of characteristic p. The action of E_{1} on [U, E_{1}] also shows that H = E_{1}C_{H}([U, E_{1}]), so H satisfies the hypothesis of 3.8. Thus there exists a B(T)- block E with H = B(T)EC_{H}(W); in particular E ≰ C^{∗}(G, T). As W_{E} ≤ W, we get from 6.6 that E is normal in H. Since E = O^{p}(E) and O^{p}(H) = O^{p}(E_{1}) ⊴⊴ G we conclude that E is subnormal in G. But this contradicts 7.4. □

Lemma 7.6 There exists F ∈ B(T) such that F ≰ C^{∗}(G, T). Moreover, for every F ∈ B(T) with F ≰ C^{∗}(G, T) there exists E ∈ B_{∗}(T) such that F ≤ E. In particular B_{∗}(G) ≠ ∅.

Proof. By 7.5 G is not a minimal parabolic group. Hence there exists a proper subgroup L ≤ G with T ≤ L and L ≰ C^{∗}(G, T). Then L ∈ L(T), and by 7.3 there exists F ∈ B(T) such that F ≰ C^{∗}(G, T).

Let F ≤ E ∈ B(T), where E is a maximal element of B(T). By 6.9 E ∈ B^{∗}(G), and as F ≰ C^{∗}(G, T), also E ≰ C^{∗}(G, T). Hence E ∈ B_{∗}(G). □

Lemma 7.7 Let E ∈ B_{∗}(T). Then EB(T) is contained in a unique maximal element L of L(G), and E ⊴⊴ L.

Proof. Let U be the set of all L ∈ L(G) containing EB(T). For every L ∈ U define

Σ_{L} := {E^{g} | g ∈ G, E^{g} ⊴⊴ L}.

Let L ∈ U and E^{g} ∈ Σ_{L}. Since E^{g} = O^{p}(E^{g}), the subnormality of E^{g} in L gives O_{p}(L) ≤ N_{L}(E^{g}) and thus [Ω(Z(O_{p}(L))), E^{g}] = W_{E^{g}}. Using 2.8 (e) and 2.16 J(T) ≤ N_{L}(E). Since E^{g} is a B(T^{g})-block, J(T) is conjugate to J(T^{g}) in N_{L}(E) and so B(T) ≤ N_{L}(E). Therefore:

7.7.1 Every element of Σ_{L} is a subnormal B(T)-block of L.

Now let N be the subgroup generated by all subnormal B(T)-blocks of L. By 6.12 either E is one of these B(T)-blocks or [N, E] = 1.

Assume the second case, so E ≤ C_{L}(N). Let B(T) ≤ S ∈ Syl_{p}(L). As C_{L}(N) does not contain any subnormal B(T)-block of L, we get from 7.3 that

E ≤ C_{L}(N)S ≤ C^{∗}(L, S).

In particular E ≤ C^{∗}(G, T^{g}) for S ≤ T^{g}. But then g ∈ N_{G}(B(T)) ≤ C^{∗}(G, T) and so C^{∗}(G, T) = C^{∗}(G, T^{g}). This contradicts E ∈ B_{∗}(T). We have shown that E is subnormal in L. Hence

7.7.2 E ∈ Σ_{L} for every L ∈ U.

Now let L̃ ∈ U and K ∈ ΣL̃. Suppose that K ≤ L. From 7.7.1, applied to L̃, we get that K is a B(T)-block. On the other hand, K = E^{g} for some g ∈ G, so K is also a B(T^{g})-block, and B(T) and B(T^{g}) are conjugate in N_{G}(K). This shows that K ≰ C^{∗}(G, T). Hence as above, K does not centralize all the subnormal B(T)-blocks of L, and 6.12 shows that K has to be one of these blocks. We have shown

7.7.3 Let K ∈ ΣL̃ and K ≤ L. Then K ∈ Σ_{L}.

Now [12, 6.7.3] shows that B(T)E is contained in a unique maximal element of L(G). □

Lemma 7.8 Suppose that [V, Z] = 1. Then O_{p}(E) ≤ O_{p}(G) for every E ∈ B(T) with E ≰ C^{∗}(G, T).

Proof. Observe that C_{T} (Z) = B(T), so [V, Z] = 1 implies V ≤ B(T). Pick E ∈ B(T) with E ≰ C^{∗}(G, T); in particular [V, E] ≠ 1 and W_{E} ≤ V. If E is not exceptional, then O_{p}(E) = W_{E} ≤ V, and we are done. Thus we may assume that E is exceptional. If O_{3}(E) ≤ C_{G}(V), then 6.6 shows that O_{3}(E) ≤ O_{3}(G). Hence, we may also assume that \overline{O_{3}(E)} ≠ 1, so W_{E} is the only non-central E-chief factor in C_{EB(T)}(V). Set

V^{∗} := ⟨V α | α ∈ Aut(B(T))⟩.

As no element of B(T) \ C_{B(T)}(W_{E}) acts quadratically on O_{3}(E)/Z(E), V^{∗} centralizes W_{E} and

W_{E} ≤ C_{B(T)}(V^{∗}) ≤ C_{EB(T)}(V).

It follows that [C_{B(T)}(V^{∗}), E] = W_{E} ≤ C_{B(T)}(V^{∗}). But C_{B(T)}(V^{∗}) is a non-trivial characteristic subgroup of B(T), and thus E ≤ C^{∗}(G, T), a contradiction. □

Lemma 7.9 Let E ∈ B(T) with W_{E} ≤ V and T_{E} := C_{T} (\overline{E}). Then [W_{E}, T_{E}] = 1 and [V, T_{E}, E] = 1.

Proof. Note that T_{E} normalizes W_{E}. Hence [W_{E}, T_{E}] ≤ Z(E) since W_{E}/Z(E) is an irreducible E-module. As [W_{E}, E] = W_{E}, a first application of the Three Subgroups Lemma gives [W_{E}, T_{E}] = 1. But then [V, E, T_{E}] = [W_{E}, T_{E}] = 1 and [E, T_{E}, V] = 1, and another application of the same lemma also yields [V, T_{E}, E] = 1. □

Notation 7.10 We use Definition 2.7. Recall that G ≠ C_{G}(V)N_{G}(J(T)), so J(T) ≰ C_{G}(V). Hence by 2.8 O_{T} (Ṽ) ≠ ∅, where Ṽ := V/V_{0} and V_{0} := C_{V} (O^{p}(G)). We set

Q_{T} (V) := ⟨A | A ∈ O_{T}^{∗} (V)⟩.

Moreover, set T_{0} := Z if \overline{Z} ≠ 1 and T_{0} := Q_{T} (V) if \overline{Z} = 1.

Lemma 7.11 Let E ∈ B(T) such that E ≰ C^{∗}(G, T) and ⟨E, T_{0}⟩ ≤ L ∈ L(T). Then T_{0} ≤ C_{T} (\overline{E}).

Proof. By 7.3 there exists E ≤ F ⊴⊴ L with F ∈ B(T). ¿From 3.8 we get that [F, Z] ≤ V, so [\overline{F} , \overline{Z}] = 1. Thus [\overline{F} , \overline{T}_{0}] = 1 if T_{0} = Z. Assume that \overline{Z} = 1. Then 7.8 gives O_{p}(\overline{F}) = 1. On the other hand, 2.16 implies [F, A] ≤ O_{p}(F) for A ∈ O_{T}^{∗} (V). Hence also in this case [\overline{F} , \overline{T}_{0}] = 1. This shows that [\overline{E}, \overline{T}_{0}] = 1.

□

Lemma 7.12 There exists E ∈ B_{∗}(T) such that T_{0} ≤ C_{T} (\overline{E}).

Proof. Let T ≤ P ≤ G such that |P | is minimal with P ≰ C^{∗}(G, T). Then by 2.3 P is minimal parabolic, and thus by 7.5 P ∈ L(T). Hence 7.3 gives a B(T)-block F ≤ P with F ≰ C^{∗}(G, T), in particular [\overline{F} , \overline{T}_{0}] = 1 by 7.11. According to 6.9 there exists E ∈ B_{∗}(T) with F ≤ E. By 6.7 we may assume that F and E are both symmetric and F < E. In particular p = 2 and O_{2}(F) ≤ O_{2}(E) ≤ V, so \overline{E} ≅ A_{m} and \overline{F} ≅ A_{m′}, 3 ≤ m′ < m, m′ and m odd.

Pick t ∈ B(T) such that R := [W_{E}, t] has order 2 and [\overline{F} , \overline{t}] ≠ 1, and set E_{t} := O^{2}(C_{E}(\overline{t})). Then \overline{E}_{t} ≅ A_{m−2} and also E_{t} is a B(T)-block not in C^{∗}(G, T). Moreover R ≤ W_{F}, and thus by 7.11 ⟨B(T)E_{t}, T_{0}⟩ ≤ C_{G}(R). Observe that ⟨F, E_{t}⟩ = E and that C_{G}(R) ∈ L(T). So applying 7.11 we see that T_{0} centralizes \overline{E}_{t} and so also \overline{E}. □

Lemma 7.13 \overline{T}_{0} = 1.

Proof. By way of contradiction we assume that \overline{T}_{0} ≠ 1. Recall that O_{p}(\overline{G}) = 1; so N\overline{G}(\overline{T}_{0}) is a proper subgroup of \overline{G}. We further set

T_{E} := C_{T} (\overline{E}), Q := ⟨A | A ∈ O_{T}^{∗} (Ṽ)⟩.

According to 7.12 there exists E ∈ B_{∗}(T) with T_{0} ≤ T_{E}; in particular \overline{EB(T)} ≤ N\overline{G}(\overline{T}_{0}). By 7.7 EB(T) is contained in a unique maximal subgroup H of G and E ⊴⊴ H.

7.13.1 \overline{H} is the unique maximal subgroup of \overline{G} containing \overline{EB(T)}; in particular \overline{T} ≤ N\overline{G}(\overline{T}_{0}) ≤ \overline{H}.

By 6.6 C_{G}(V) ≤ N_{G}(E) and by 7.4 N_{G}(E) ≤ H; so \overline{H} is a maximal subgroup of \overline{G}, and the uniqueness property of H implies that of \overline{H}. As \overline{EB(T)} ≤ N\overline{G}(\overline{T}_{0}) and \overline{T}_{0} is normal in \overline{T}, we also get the additional assertion.

7.13.2 [E, Q] ≤ O_{p}(E), and N_{G}(Q) ≤ H if Q_{T} (V) ≠ 1.

Note that C_{H}(Ṽ) = C_{H}(V) because O_{p}(\overline{G}) = 1. Then 2.16, applied to \overline{H} and Ṽ, shows that [E, Q] ≤ O_{p}(E). On the other hand, Q is normal in N_{G}(QO_{p}(E)), so EB(T) ≤ N_{G}(Q). The uniqueness of H gives either N_{G}(Q) ≤ H or \overline{Q} = 1. In the second case 2.8 implies that also Q_{T} (V) = 1.

7.13.3 N_{G}(\overline{T}_{1}) ≤ \overline{H} for every \overline{B(T)}-invariant subgroup 1 ≠ \overline{T}_{1} ≤ \overline{T}_{E}.

As O_{p}(\overline{G}) = 1, N\overline{G}(\overline{T}_{1}) is a proper subgroup of \overline{G} containing \overline{EB(T)}. Hence 7.13.1 implies N_{G}(\overline{T}_{1}) ≤ \overline{H}.

According to 7.5 G is not minimal parabolic. Thus there exists a proper subgroup P ≤ G with T ≤ P and P ≰ H. We choose P such that |P | is minimal with that property. Then P is minimal parabolic since N_{G}(T) ≤ H. Observe that G = ⟨P, E⟩ by the uniqueness of H. Set A := Z ∩ O_{p}(P) and S := ⟨A^{P} ⟩.

7.13.4 Either \overline{A} = 1 or [W_{E}, S] ≠ 1.

Recall that E has a unique non-central chief factor in V. Assume that [W_{E}, S] = 1. Then C_{V} (S) is P - and B(T)E-invariant, so C_{V} (S) is G-invariant. But now the definition of V shows that V = C_{V} (S) and \overline{S} = 1.

7.13.5 \overline{S} = 1.

Assume that \overline{S} ≠ 1. Then T_{0} = Z and according to 7.13.4 there exists y ∈ P such that [W_{E}, Y] ≠ 1 for Y := A^{y}. If Y normalizes W_{E}, then by 7.3 Y also normalizes \overline{E}, and [W_{E}, E] = W_{E} implies that [W_{E}, Y, E] ≠ 1. The action of B(T) on W_{E} shows that [W_{E}, Y] ∩ Z ≰ Z(G). If Y does not normalize W_{E}, then by 7.3 Y also does not normalize (W_{E} ∩ Z)C_{V} (G) and [W_{E} ∩ Z, Y] ≰ Z(G).

Hence in both cases R := [V, Y] ∩ Z ≰ Z(G), so C\overline{G}(R) is a proper subgroup of \overline{G}. On the other hand, by 7.11

R ≤ [V, Y] ≤ [V, T_{E}^{y}],

and so by 7.9 [E^{y}, R] = 1. Thus also [B(T)^{y}E^{y}, R] = 1 since R ≤ Y ≤ Z^{y}. The uniqueness of H^{y} implies

B(T) ≤ C_{G}(R) ≤ H^{y}.

In particular B(T) and B(T)^{y}, and thus also Z and Z^{y} are conjugate in H^{y}. It follows from 7.13.1 that

\overline{EB(T)} ≤ N\overline{G}(\overline{Z}) ≤ \overline{H}^{y}.

The uniqueness of H yields H = H^{y} and y ∈ H. Now 6.12 shows that E is also an B(T)^{y}-block and by 7.9 and 7.11 [W_{E}, Y] = 1, contradicting the choice of Y.

Let W_{0} := C_{V} (O^{p}(P)) and choose 1 ≠ W ≤ V minimal such that W = [W, O^{p}(P)]. Then U := W/W ∩ W_{0} is an irreducible P -module. Observe that by 3.3

C_{T} (W) ≤ C_{T} (U) = O_{p}(P).

7.13.6 \overline{Z} ≠ 1, so T_{0} = Z.

Assume that \overline{Z} = 1. Then T_{0} = Q_{T} (V), so O_{T}^{∗} (V) ≠ ∅ and thus by 2.8(c) also O_{T}^{∗} (Ṽ) ≠ ∅. Moreover, by 7.13.2 and 7.8 Q ≤ T_{E} and by 7.9 [V, Q, E] = 1. This shows that G = ⟨E, P ⟩ ≤ N_{G}(O_{O_{p}(P)}^{∗}(Ṽ)) and so

O_{O_{p}(P)}^{∗}(Ṽ) = ∅.

Let Q_{0} := Q∩O_{p}(P), and W_{1} := [W, Q_{0}]. Then [W_{1}, O^{p}(P)] = 1 and by 7.9 [W_{1}, E] = 1, so W_{1} ≤ V_{0} and W̃_{1} = 1. Furthermore, let A ∈ O_{T}^{∗} (Ṽ) and A_{0} := A ∩ O_{p}(P). Then 2.8 (b) implies that |A/A_{0}| > |W̃/CW̃(A)|, and thus also

|A/C_{A}(U)| > |U/C_{U}(A)|,

since C_{T} (U) = O_{p}(P) and C_{A}(U) = A_{0}. On the other hand, by 5.6 and 3.6 applied to P/C_{P} (U) we get |A/C_{A}(U)| = |U/C_{U}(A)|, a contradiction.

7.13.7 [W, O_{p}(P)] = 1.

By 7.13.5 and 7.13.6 [\overline{O_{p}(P)}, \overline{Z}] = 1 and \overline{Z} ≰ \overline{O_{p}(P)}, so [O_{p}(P), O^{p}(P)] ≤ C_{P} (W) using 3.3. The Three Subgroups Lemma gives [W, O_{p}(P)] = 1, since W = [W, O^{p}(P)].

We now derive a final contradiction using 7.13.6 and 7.13.7. From 3.3 we get that C_{T} (U) = C_{T} (W) = O_{p}(P); in particular O_{p}(P/C_{P} (W)) = 1. Hence again 5.6 and 3.6 imply

(∗)  |A/C_{A}(W)||C_{W} (A)| = |W | for A ∈ O_{P} (W).

If [W, J(T)] = 1, then Z ≤ J(T) ≤ O_{p}(P), which contradicts 7.13.5. Thus we have [W, J(T)] ≠ 1. But now an elementary argument using (∗) gives

(A ∩ O_{p}(P))W ∈ A(T) for every A ∈ A(T),

so W ≤ J(T) and Z ≤ C_{T} (W) = O_{p}(P), again a contradiction to 7.13.5. □

Lemma 7.14 Let E ∈ B_{∗}(T). Then there exists A ≤ B(T) such that the following hold:

(a) [E, A] = E and [V, A, A] = 1.

(b) |V/C_{V} (A)| = |\overline{A}| and C_{V} (A) = C_{V} (a) for every a ∈ A \ C_{A}(V).

Proof. If E is a symmetric block we let F ≤ E be the B(T)-block given by

6.8 with F/O_{p}(F) ≅ SL_{2}(2)′ and otherwise set F := E. Thus in all cases F is a linear block not in C^{∗}(G, T). Hence 7.8 and 7.13 give O_{p}(F) ≤ O_{p}(G). The action of F on W_{F} shows that

(1)  |B(T)/C_{B(T)}(W_{F})| = q.

Observe that by 7.13 [V, Z] = 1, so V ≤ B(T). Set

W^{∗} := ⟨W_{F} α | α ∈ Aut(B(T))⟩ and V^{∗} := ⟨V α | α ∈ Aut(B(T))⟩.

Assume first that F is not exceptional. Then

6.3 implies [O_{p}(G), F] ≤ W_{F} ≤ W^{∗}. As F ≰ C^{∗}(G, T), this shows that W^{∗} ≰ O_{p}(G). Hence there exists α ∈ Aut(B(T)) such that A := W_{F} α ≰ O_{p}(G); in particular A ≰ O_{p}(F) and [E, A] = E. The action of A on W_{F} and (1) give

(2)  q = |W_{F} /C_{W_{F}} (A)| = |V/C_{V} (A)| and [V, A, A] = 1.

As W_{F} /C_{W_{F}} (F) is a 2-dimensional SL_{2}(q)-module, we also get

(3)  C_{V} (A) = C_{V} (a) for every a ∈ A \ C_{A}(V).

Hence, A satisfies (a) and (b).

Assume now that F is exceptional; so F = E. Then no element in B(T)\C_{B(T)}(W_{E}) acts quadratically on O_{3}(E)/Z(E). It follows that W^{∗} is elementary abelian and [V^{∗}, W_{E}] = 1. By 3.7 O_{3}(E) ≤ B(T), so there exists α ∈ Aut(B(T)) with A := O_{3}(E)α ≰ C_{B(T)}(E/O_{3}(E)) for otherwise ⟨O_{3}(E)α | α ∈ Aut(B(T))⟩ is a characteristic subgroup of B(T) normalized by E, contradicting E ≰ C^{∗}(G, T). In particular [E, A] = E.

Observe that V^{∗} ≤ C_{B(T)}(W_{E}α) ∩ C_{B(T)}(W_{E}), so

[V^{∗}, O_{3}(E)] ≤ Ω(Z(O_{3}(E))) and [E, C_{V} (E)α^{−1}] ≤ O_{3}(E).

This shows that

[O_{3}(E), O_{3}(E)C_{V} (E)α^{−1}] = Z(E)[O_{3}(E), C_{V} (E)α^{−1}] ≤ [O_{3}(E), C_{V} (E)α^{−1}]Z(B(T)).

As [O_{3}(E), O_{3}(E)C_{V} (E)α^{−1}] is an E-submodule of Ω(Z(O_{3}(E))), we get that either

W_{E} ≤ [O_{3}(E), C_{V} (E)α^{−1}]Z(B(T)) or [O_{3}(E), C_{V} (E)α^{−1}] ≤ Z(E).

In the first case W_{E}α ≤ [A, C_{V} (E)]Z(B(T)) ≤ C_{V} (E)Z(B(T)) and thus O_{3}(E) ≤ C_{B(T)}(W_{E}α). But then

[A, O_{3}(E)] ≤ Z(A) and [O_{3}(E), A, A] = 1,

which contradicts the definition of an exceptional B(T)-block.

So we are in the second case, in particular

[O_{3}(E), C_{V} (E)α^{−1}, E] = 1 and [E, O_{3}(E), C_{V} (E)α^{−1}] = [O_{3}(E), C_{V} (E)α^{−1}].

Observe that either [C_{V} (E)α^{−1}, E] = O_{3}(E) or [C_{V} (E)α^{−1}, E] ≤ Z(O_{3}(E)). Hence the Three Subgroups Lemma gives

[O_{3}(E), C_{V} (E)α^{−1}] = Z(E) or [O_{3}(E), C_{V} (E)α^{−1}] = 1, respectively.

Assume that [O_{3}(E), C_{V} (E)α^{−1}] = Z(E). Then Z(E)α = [A, C_{V} (E)] ≤ C_{V} (E) and thus

[O_{3}(E), A, A] = [O_{3}(E) ∩ A, A] ≤ Z(E)α ≤ C_{V} (E) ∩ O_{p}(E) = Z(E).

Now A acts quadratically on O_{3}(E)/Z(E), which contradicts the definition of an exceptional B(T)-block.

Thus, we have [O_{3}(E), C_{V} (E)α^{−1}] = 1 and so [A, C_{V} (E)] = 1. Now as above (2) and (3) hold for A, so A satisfies (a) and (b). □

Theorem 7.15 No group satisfies Hypothesis 7.1.

Proof. Let Y be the set of all subgroups A ≤ B(T) for which there exists E ∈ B_{∗}(T) such that A and E satisfy (a) and (b) of 7.14, and let

D := ∪_{g∈G}Y^{g} and \overline{D} := \{\overline{A} | A ∈ D\}.

We will show that \overline{D} satisfies Hypothesis 4.3.

It is evident from 7.14 that \overline{D} satisfies (i) and (ii) of 4.1. Moreover, 7.13 shows that property (∗∗) of 4.3 holds. Next we prove (iii) of 4.1.

Let A, B ∈ D such that [\overline{A}, \overline{B}] = 1. If C_{V} (A) = C_{V} (B), then 4.3 (∗∗) yields \overline{A} = \overline{B}. Assume that C_{V} (A) ≠ C_{V} (B). Then by 4.1 (ii) \overline{A}∩\overline{B} = 1 and so |\overline{A}\overline{B}| = |\overline{A}||\overline{B}|. On the other hand, by 4.1 (ii) |V/C_{V} (AB)| ≤ |\overline{A}||\overline{B}|, so again (∗∗) gives |\overline{A}\overline{B}||C_{V} (AB)| = |V |. This proves (iii) of 4.1.

Finally, we show property (∗) of 4.2 with M := \overline{C^{∗}(G, T)}. Let L := N_{G}(D ∩ T) and recall that C_{G}(V) ≤ C^{∗}(G, T). Hence

N\overline{G}(\overline{D} ∩ \overline{T}) ≤ M ⟺ L ≤ C^{∗}(G, T),

so we may assume by way of contradiction that L ≰ C^{∗}(G, T). Then L ∈ L(T), and by 7.3 there exists a B(T)-block E in L which is not in C^{∗}(G, T). According to 7.6 E ≤ F ∈ B_{∗}(T). But then by 7.14 there exists A ∈ D ∩ B(T) such that [F, A] = F. This contradicts A ∈ D ∩ T and F ≤ L.

We have shown that \overline{D} satisfies Hypothesis 4.3. By 4.18 there exists a subnormal subgroup E^{∗} in G such that C_{G}(V) ≤ E^{∗} ≰ C^{∗}(G, T) and \overline{E}^{∗} satisfies (c) and (d) of 4.18. Moreover, by the definition of D and 4.18 (a) there exists E ∈ B_{∗}(G) and A ∈ D such that EA ≤ E^{∗}; in particular B(T) normalizes E^{∗}. Now 3.8, applied to B(T)E^{∗}, together with 6.6 shows that E is normal in E^{∗}, and thus subnormal in G. This contradicts 7.4. □

The Proof of Corollary 1.9: Let M be the unique maximal subgroup of G containing T. As every characteristic subgroup X of B(T) is also characteristic in T, we get T ≤ N_{G}(X). Hence N_{G}(X) ≤ M if X is non-trivial. Similarly C_{G}(Ω(Z(T))) ≤ M. It follows that C^{∗}(G, T) ≤ M; in particular C^{∗}(G, T) ≠ G. Hence G satisfies the hypothesis of the Local C^{∗}(G, T)-Theorem and the Local C^{∗∗}(G, T)-Theorem for Minimal Parabolic Groups. In particular, for every subnormal symmetric B(T)-block E not in C^{∗}(G, T), E/O_{2}(E) ≅ A_{2^{n}+1}. Thus, G satisfies the conclusion of the Local C(G, T)-Theorem. Moreover, (e) of the Local C^{∗}(G, T)-Theorem together with the fact that G is a minimal parabolic gives the additional statement in the conclusion of 1.9.

Acknowledgement. We would like to thank U. Meierfrankenfeld and Ch. Parker for reading an earlier draft of the manuscript and for their helpful suggestions.

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