13.7. FINISHING THE TREATMENT OF A 6 WHEN (VGi) IS NONABELIAN 937K := 02 (K1), where Ki is a minimal normal subgroup of H* contained in Op(C*);
thus K* ~ K/02(K) ~ Ep"'" n := 1or2 since His an SQTK-group. The subgroupK satisfies one of these two hypotheses throughout the rest of the proof.
In either case, K = 02 (K) is subnormal in H, so 02 (K) ::::; QH and QH
normalizes K. As K::::; C, [UH, K]::::; U 0 , so
(!)
Then 1 =f. [Uo,K]::::; [fh(Z(02(K))),K], so that KE Xf.
Consider for the moment the case where KE C(H). Then KE £1(G,T), sothat K* ~ K/02(K) is described in the list of 13.5.2.1, and K :::'.] H by 13.3.2.2. We
saw L1 i:. C, so L1 i:. K. Thus K/0 2 (K) is not A 6 or A 6 by A.3.18, so K* ~ Lg(2)
or A5 by 13.5.2.1. Then as L1 = [L1,T], either [K,L1]::::; 02(K), or K* ~ A 5 and
K = [K, L1]. Further if K* is A5, then by 13.3.2.3, each IE Irr +(K, R2(KT), T)
is a T-invariant A5-module.We now return to consideration of both cases. By the previous paragraph we
have K :::'.] H. Since K* is either simple or a p-group for p odd, and CK(UH) =
02 (K) ::::; CK(Uo) by (!),we conclude from Coprime Action that
Observe that if K::::; M, then K normalizes V by 13.7.3.9, so (V3, K]::::; VnU 0 =
V 1 , and hence k =f. L 1 , contrary to 13.3.9 applied to K in the role of "Y". Thus
Ki:. M.
Suppose next that Lis an A 6 -block. Then L 1 has just two noncentral 2-chief
factors, while L1 is nontrivial on ViUo/Uo and hence also nontrivial on QH /He
by (2). Therefore L 1 centralizes Uo, and hence [K, L1] ::::; CK(Uo) = 02(K) using
(*), so K acts on 02 (L 102 (K)) = L1. Then as Vi ::::; L 1 and K ::::; C, [Vi, K] ::::;
02(L1) n Uo ::::; Z(L)Vi, so K centralizes V3 by Coprime Action as IZ(L)I ::::; 2 by
C.1.13.b. Thus K::::; G 1 n G 3 ::::; Mv by 13.5.5, whereas we saw Ki:. M.
Therefore L is not an A 6 -block. Then by 13.2.2.7, Nc(B) ::::; M, where B :=
Baum(R 1 ). In particular as Ki:. M, Bis not normalized by K.
Assume next that K* ~ A5 and K = [K, L1]. Then Ri =(Kn T)02(KR1),
and we saw earlier that each I E Irr +(K, R2(KT), T) is an A5-module, so J(R1)
centralizes I by B.4.2. Then B ::;! KT by B.2.3.5, contrary to the previous para-
graph. This contradiction shows that [K, L 1 ] ::::; 02 (K) in the case that KE C(H).
. Next consider the case where K* is a p-group. As L1 acts on 02(K), 02(K) ::::;
Ri, so Ri E 8yl2(KR1). As CK(Uo) = 02(K) by(*), CKR 1 (R2(KR1)) = 02(KR1)
by A.1.19. Thus if J(R 1 ) centralizes R2(KR1), then B ::;! KR1 using B.2.3.5,
whereas we saw Bis not normalized by K. Thus KR1 satisfies case (2) of Solvable
Thompson Factorization B.2.16; so in particular p = 3. Since K* is a minimal nor-
mal subgroup of H, Tis irreducible on K, so by B.2.16.2, J(KR 1 )/02(J(KR 1 )) ~
83 or 83 x 83. As L 1 i:. C acts on J(KR1), the latter case is impossible as
m 3 (H)::::; 2. Thus if K is a p-group, we conclude K ~ Z3.
We have now shown that K ~ Z3, A5, or L3(2), and that L1 centralizes K.
In particular, K acts on 02 (0 2 (K)L 1 ) = L1. Let U::::; U 0 be minimal subject to
U ::;! X := KL 1 T and [U, K] =f. 1. Set x+ := X/Cx(U); we claim that 02(X+) = 1:
For 02 (K+) = 1 as 02(K) centralizes Uo, so K+ centralizes 02(X+), and hence
02 (X+) = 1 by the Thompson Ax B Lemma and mininality of U, as claimed. As