i4.4. FINISHING THE TREATMENT OF (VG1) NONABELIAN ioo5Let Xi be of order 3 in Li. Then Q = [Q, Xi]CQ(Xi) with [Q, Xi] = [U, Xi] ~
Qg by 14.3.32. Now if Qi is the preimage of an irreducible Xi-submodule of [Q,°Xi],
then by 12.8.4.2, Cq(Xi) normalizes Qi; further Cq 1 (CQ(Xi)) >Vi= Cq 1 (Xi) by
the Thompson Ax B-Lemma, so Cq(Xi) centralizes Qi as Xi is irreducible on Qi.Thus Cq(Xi) centralizes [Q,X1] = [U,X1], so Zu = CT(U) = Cq(X1)nCq(Zu) is
of index at most 2 in CQ(X1) as Zu ~ Z4. Thus either Cq(X 1 ) = Zu, or Cq(X 1 )
is dihedral or quaternion of order 8.
Suppose first that CQ(Xi) = Zu. Then Q = U, so as H =KT, JHJ 2 = 29 by
14.3.32. Hence as we saw M =LT, M = L using (1), so (3) holds.
So we assume Cq(X1) is of order 8, and it remains to derive a contradic-
tion.1 Now CQ(X1) :::; 02(LT), so 02(LT) = PCQ(X 1 ). Then as M = LT,
M = LCq(X1).
For r E Cq(X1) - U, r centralizes the supplement [U, X1] to Pin 02 (L 1 ), so
from the structure of Aut(L3(2)), r centralizes L/P. Then by Gaschiitz's Theorem
A.1.39, we may chooser so that [r, L] :::; V. Now as Lis irreducible on V, r is an
involution, and as CT(L) = 1, P induces the full group of transvections on V(r)
with axis V. So L = PCL(r) by a Frattini Argument, and r inverts P.
Let TL := T n L, so that TL is of index 2 in T. As G is simple, ThompsonTransfer says there is g E G with r^9 E TL. We show that any such rB is not extremal
in M; then the standard transfer result Exercise 13.1 in [Asc86a] contradicts r E
02 (G).
As H contains no L 3 (2)-section, r^0 n V = 0. Thus rB E TL - P as V =
01 (P), and conjugating in L, we may take r^9 E 02(L1) =PU. By 14.3.32.2, eachnontrivial coset of Zu in U contains exactly two involutions fused under U, and
by 14.3.32.1, K is transitive on fj#, so K is transitive on involutions in U - V 1.
Thus as r^0 n V = 0, rB f:. U. Then as P_ E Syl 2 (K), V = C 0 (r9); so as
PU centralizes Zu, Cu(rB) = ZuCv(r^9 ) = ZuVi- Thus JU : Cu(r9)J = 23 , so
JCT(rB)J:::; 27 as ITI = 210. Therefore ifrB is extremal in M, then CT(rB) = CT(r)9.
As V is the natural module for CL(r)/V ~ L3(2), Vi = Z(CT(r)) n (CT(r)).
Then as Vi :::; Z(CT(r^9 )) n (CT(rB)), we conclude g E H. This is impossible
as r E Q = 02 (H), while r9 E PU but rB f:. U = Q n PU. This contradiction
completes the proof of (3), and hence of 14.3.33. D
At this point, we can complete the identification of Gas HS, and hence estab-
lish Theorem 14.3.26. Namely by 14.3.33 and 14.3.32, U = Q = 02(H) ~ Z4 Qg
with H/U ~ 85 • By 14.3.32.4, U is an indecomposable module under the action of
H. Further by 14.3.33, F(M) = P ~ Z~, and M/P ~ L 3 (2). Thus G is of type
HS in the sense of section I.4 of Volume I, so we quote the classification theorem
stated there as I.4.8 to conclude that G ~HS.
13.7. Finishing the treatment of A 6 when (VG^1 ) is nonabelian
In this section, we assume Hypothesis 14.3.1 holds, and continue the notation
of section 14.3. In addition, we assume U := (V^01 ) is extraspecial. In particular,
Hypothesis 14.3.10 holds, and we can appeal to results in the later subsections of
section 14.3.
(^1) Notice we are here eliminating the shadow of Aut(HS).