1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

12.3. ELIMINATING A7 807

By 12.2.26, the structure of L, and hence also of Ca(z), are determined, so we

can move toward the identification G using recognition theorems from our Back-

ground References.

PROPOSITION 12.2.27. ( 1) If n = 4, then G ~ M 24 or L 5 (2).

(2) If n = 3 then G ~ L4(2) or Ag.

PROOF. Let z E v n z#. By assumption, Ca(z) ~ M, so we conclude from

12.2.25.1 that Ca(z) = CM(z) = CL(z). By 12.2.26, L is determined up to iso-

morphism, so as Ln+l (2) satisfies the hypotheses on G, Ca(z) is isomorphic to the

centralizer of a transvection in Ln+i(2). Hence if n = 4 then by Theorem 41.6 in

[Asc94], G ~ M24 or L5(2). Similarly if n = 3 then G ~ L 4 (2) or Ag by I.4.6. D

By 12.2.20, L/02(L) ~ Ln(2), and by 12.2.25.2, n = 3 or 4. Thus one of the

conclusions of Theorem 12.2.13 holds by 12.2.27. Therefore the proof of Theorem

12.2.13 is complete.

12.3. Eliminating A7

In section 12.3 we eliminate the cases where LE Lj(G,T) with L/0 2 ,z(L) ~

A1; namely we prove:

THEOREM 12.3.1. Assume Hypothesis 12.2.3. Then L/0 2 ,z(L) is not A1.
We adopt the conventions of Notation 12.2.5, including Z = Q 1 (Z(T)).
After Theorem 12.3.1 is established, case (ii) of 12.2.7.3 cannot arise, so we
obtain:

COROLLARY 12.3.2. Assume Hypothesis 12.2.3, and further assume Ca(Z) ~

M. Let HE 1-l*(T,M) and set K := 02 (H) and VK := (ZK). Then either

(1) H is solvable, or

(2) K/02(K) ~ L2(4), KE Lj(G,T), and [VK,K] is the sum of at most two

A5-modules for K/02(K).

We mention some shadows which the analysis must at least implicitly handle:

As we noted in Remark 12.2.14, in the QTKE-group G = M 23 there is LE Lj(G, T)
with L ~Ad E 16 • The case G = M 23 is explicitly excluded by Hypothesis 12.2.3,

and its shadow is eliminated early in this section by an appeal to Theorem 12.2.13.

The group G = M cL is quasithin but not of even characteristic, in view of

the involution centralizer isomorphic to A 8 ; this group has L E .C*(G, T) with

L ~ Ad E 16. Further G = 07 (3) is not quasithin but has L E £*(G, T) with

L ~ A1 / E 64. The shadows of these two groups are eliminated by control of the

centralizer of a 2-central element of V whose centralizer in the shadow is not in 7-le.

In the remainder of this section we assume G, L, M afford a counterexample to

Theorem 12.3.1. Choose a V E Irr+ ( L, R 2 (LT), T); then V is described in 12.2.2.3.
We now begin a series of reductions.
LEMMA 12.3.3. V is the natural permutation module of rank 6 for L ~ A1.

PROOF. By 12.2.2.3, either V/Cv(L) is the natural module for Lor m(V) = 4.

In the first case since V = [V, L] and the 1-cohomology of the natural module is

trivial by I.1.6, the lemma holds.