798 12. LARGER GROUPS OVER Fz IN .Cj(G, T)
M = Na(X 0 ) since Na(Xo) = !M(XoT) by 1.3.7, so H::::; YT::::; Na(Xo) ::::; M,
contr,adicting H i. M. D
Given a group A, write B(A) for the subgroup of A generated by all elementsof order 3 in A.
LEMMA 12.2.8. One of the following holds:
(1) 031 (M) = L.
(2) L/02(L) ~ A5 or L3(2).
(3) L/0 2 (L) ~ A 6 or A 7 and L = B(M) is the subgroup of M generated by all
elements of M of order 3.
PROOF. First Lis described in 12.2.2.3, so if m3(L) = 1, then (2) holds; thus
we may assume m 3 (L) = 2. Then (1) or (3) holds by 12.2.2 and A.3.18. D
LEMMA 12.2.9. (1) If Cz(L)-:/= 1, then Ca(Z) ::::; M.
(2) If Ca(Z)::::; M, then L = [L, J(T)].PROOF. As M = !M(LT), (1) holds. Theorem 3.1.8.3 implies (2). D
LEMMA 12.2.10. (1) CMv(L) = Z(L).
(2) L = 02 (Mv) and Mv =LT.
PROOF. In each case listed in conclusion (3) of Theorem 12.2.2, Out(L/0 2 (L))is a 2-group, so 02 (Mv) ::::; LCMv(L). Further in cases (a)-(f), the irreducible
module I:= V/Cv(L) satisfies E := EndI,(I) ~ F 2 , so that CMv(L) = 1. Hence
(1) and (2) hold in these cases. In case (g), I= V and E ~ F 4 , with Z(L) inducing
E#, so again (1) and (2) follow. D
LEMMA 12.2.11. Assume HE H*(T, M) with H::::; Na(U) for some 1-:/= U::::;
V. Assume also that one of the following holds:
(a) L/02(L) ~ L5(2).
(b) L/02(L) ~ A5, and V::::; 02(Ca(v)) for v E Cv(T)#.
(c) L/02(L) ~ G2(2)' and Ca(Vi) ::::; M, where Vi is the (T n L)-invariant
subspace of V of rank 3.
Then(1) n(H) ::::; 2, and
(2) if n(H) = 2, then a Hall 2' -subgroup of H n M is a nontrivial 3-group.
PROOF. The lemma is vacuously true if n(H) ::::; 1, so we may assume that
n(H) ~ 2. Then by E.2.2, H n M is the preimage of the normalizer of a Borel
subgroup of the group 02 (H/0 2 (H)) of Lie type and characteristic 2. We take C
to be a Hall 2'-subgroup of H n M, so that C is abelian and CT= TC. We may
assume that either:
(I) n(H) = 2, but there is a prime p > 3 such that B := Op(C)-:/= 1, or
(II) n(H) > 2, in which case p and B also must exist.
Then also A := fh(B) -:/= 1, BT = TB, and AT= TA. As n(H) > 1 and
AT= TA, NH(A) i. M by 4.4.13.1.
Next as B ::::; H n M ::::; NM(U) by hypothesis, B normalizes V by 12.2.6.
We claim in fact that B centralizes V: For otherwise 1 -:/= B ::::; 02 (Mv) = L by
12.2.10.2. Thus Bis an abelian p-subgroup of L with p > 3, and BT= TB, so we