826 12. LARGER GROUPS OVER F 2 IN .Cj (G, T)whereas (vo) = Gvv (Lv) = Gvv (Kv) with Vv/ (vo) of rank 4 and self-centralizing inAut(Kv). This contradiction completes the proof. D
Set U := 01(Z(02(KvS))). Recall (KvS)* = KvS/02(KvS).
LEMMA 12.6.13. (1) F*(KvS) = 02(KvS) = GKvs(U).
(2) K; is simple.
(3) V,,::; [U,Kv]·PROOF. By 12.6.12, Kv is not quasisimple, while by 12.6.3, K; is quasisimple.
Since Lv/0 2 (Lv) has trivial center and contains an E 9 -subgroup, if 02(Kv) <
02,3(Kv) then m3(Lv02,3(Kv)) = 3, contrary to Gv an SQTK-group. Therefore
from the list of possibilities in 1.2.1.4b, K; is simple, so F*(KvS) = 02(KvS) asKv is not quasisimple. We showed V,, E R.2(Lv), so that Lv E Xf by A.4.11. By
12.6.4 and 12.6.5, 02(KvS) ::; R ::; Ns(Lv) E Syl2(NaJLv)), so we may apply
A.4.10.3 with Lv, Kv, S in the roles of "X, Y, T" to conclude that Kv E Xi;
then [R2(KvS), Kv] f 1 by A.4.11. As K; is simple, U = R2(KvS) and Gs(U) =
02(KvS). Similarly by A.4.10.2, Vv::; [U,Kv]· D
LEMMA 12.6.14. J(Tv) f;. Q.
PROOF. Assume J(Tv) :S Q. By 12.6.8, S = Tv, R = Rv, and J(S) = J(R) =J(Q), so that Na(R) ::; Na(J(R)) ::; M = !M(LT). By 12.6.4, 02(KvS) ::; R.
Thus NK::;(R) = NKv(R) ::; M:. Then as Kv f;. M by 12.6.11, it follows that
R* f 1. Since Tv E Syl2(Gv), by 1.2.4 the embedding Lv < Kv is described in
in A.3.12; so as R* f 1, we conclude that K: ~ M22 or M23. Then K; has noFF-module by B.4.2, so that J(S) ::; Gs(U) by B.2.7. But Gs(U) = 02 (KvS)
by 12.6.13.1, so Kv ::; Na(J(S)) ::; M, contrary to Kv f;. M. This contradiction
completes the proof of 12.6.14. DLEMMA 12.6.15. U is an FF-module for K;S*.
PROOF. By 12.6.14 and 12.6.9, Lv = [Lv, J(Tv))]. Thus Kv = [Kv, J(Tv)],
so rv,A• ::; 1 for some A E A(Tv) by B.2.4.1, and hence U is an FF-module forK;S*. D
LEMMA 12.6.16. Assume Zv f 1 and SE Syl 2 (G). Then
(1) Na(Rv) f;. M, and
(2) L is not a block.
PROOF. By 12.6.14 and 12.6.9, L = [L, J(T)]; so as Zv f 1 by hypothesis, L
centralizes Z by 12.6.1.6. Therefore Ga(z) ::; M = !M(LT) for each z E z#.
As S E Syl2 ( G) by hypothesis, v is 2-central in G, so there is g E G with SB = Tand hence vB E Z. Further L£ < KZ by 12.6.11, so as G£ ::; M, KZ ::; 031 (M) = L
by 12.6.1.5. Also L ::; Ga(Z) ::; G£, and KZ = 031 (G£^00 ) by 12.6.3, so KZ = L.
Thus L£ ::; L, and L£ is normal in the preimage of L£ in L by 12.6.4. Hence as L
is transitive on its subgroups isomorphic to A 6 ,· L£ EL~; then as L centralizes Z,
without loss L£ = Lv. Then R£ E R1;/LT(Lv), so we also take R£ = Rv.
Thus if Na(Rv)::; M, then g E Na(Rv)::; M = Na(L), so KZ = L =LB, and
hence Kv = L::; M, contrary to 12.6.11. Therefore (1) is established.
As Na(Rv) f;. M ~ Na(Q), Q < Rv. Then as we saw at the start of the
proof of Theorem 12.6.2, Rv = ((1, 2)), so that LRv =LT. As (KvS)B =LT and
L£ = Lv, R = 02(NLT(Lv)) = Rv, and hence RB = R£ = Rv = R. Since it only