i3.5. THE TREATMENT OF A5 AND Aa WHEN (v;^1 ) IS NONABELIAN 923
LEMMA 13.5.20. V* centralizes F(H*).PROOF. If [Op(H), V] =/= 1 for some prime p, then by 13.5.13.2, we may apply
F.9.5.6 to conclude that p = 3. Let P be a supercritical subgroup of 03 (H).
Then [P, V] =/= 1, and m(P) ~ 2 since H = H/QH is an SQTK-group. Define
Y and Hi as in 13.5.19. By definition Y normalizes V, so as V is of order 2, Y
centralizes V*. Suppose 02(Y*) centralizes P*. Then as Y* /0 2 (Y*) is of order 3,
the Thompson Ax B Lemma shows that [ C p• (Y*), V*] =/= 1. This is a contradictionas Gp• (Y*) ~Hi by 13.5.19.1, and V ~ 02(Hi) by 13.5.19.2.
Therefore 02(Y*) is nontrivial on P*; then as Y = 03 ' (Y), P* is not cyclic,
so using A.1.25, P* ~ E 9 or 31+^2 and Y/Cy(P* /'P(P*)) ~ 8L 2 (3). In particularY is irreducible on P /if!(P), so as [Y, V] = 1, V = Z(Y) inverts P /if!(P).
However by F.9.5.2, m([UH, V*]) = 2; since a faithful irreducible for 8L 2 (3)/Eg is
of rank 8 and the commutator space of Z(8L 2 (3)) on such a module is of rank 4,
we conclude P* ~ 31+^2. But now X of order 3 in Y centralizes an E 9 -subgroup of
P, contradicting m 3 (H) ~ 2. DBy 13.5.20, [K, V] =/= 1 for some KE C(H) with K* ~ K/0 2 (K) quasisimple.
Let K have this meaning for the remainder of the section.
LEMMA 13.5.21. {1) K = [K, V*] and Li ~ K.
{2) KV ~ 86, As, or G2(2)'.
{3) UH= [UH,K], and UH/Co-H(K) is the natural module for K.
(4) If n = 6, then L/02(L) ~ A6 rather than A.6.
(5) K :SI H, LiT is the stabilizer in KT of the 2-subspace V3 = [UH, V]
of UH, and UH= [QH,K].
{6) UH= [UH, Li] ..
PROOF. As V has order 2 and V :SI T, V ~ Z(T). Therefore V centralizes
(T n K) E 8yl2(K), and hence normalizes K* by 1.2.1.3, as does Li = 02 (Li)
by that result. Then as [K, V] =/= 1 by choice of K, K = [K, V*], establishing
the first part of ( 1).
Define Y ~ Li as in 13.5.19. As K = [K, V], K 1. Hi by 13.5.19.2, so Y*
does not centralize K by 13.5.19.1. Therefore K = [K, Y] as K* is quasisimple.
Then since Y ~Li, K = [K,Li].
Let Tx := NT(K), X := KLiTx, and X := X/Cx(K). As Tx E 8yl2(NH(K))
by 1.2.1.3, Tx E 8yb(X). As K is quasisimple, F*(X) = k is simple. We claim:
(i) V is generated by an involution in the center of the Sylow 2-subgroup Tx
of X.
(ii) 1 =!= Y ~ti ~ 02,3(0_x(V)).
(iii) [UH, V] = V3 = [V3, Li] is of rank 2.
(iv) If (V, vx) is not a 2-group, then (V, vx) ~ 83.
Part (i) follows as V ~ Z(T) and V is of order 2 and faithful on K. Part (iii)
follows from F.9.5.2, and (iv) is a consequence of F.9.5.6.2. As K* is quasisimple
with 02(K) = 1, Ck(V) = o;:(V). Further by F.9.5.3, CK·(V) = NK(V).
Then as Li :SI H n Mv = NH(V), Ck(V) acts on Li. Thus as LiT acts on Li
and O_x(V) = Ck(V)LiTx, and as we saw earlier that K = [K, Y*], we conclude