820 .12. LARGER GROUPS OVER F2 IN £.j(a,T)by E.6.12; then as U is normal in some Sylow 2-subgroup of LT, E.6.13 supplies a
contradiction. DLEMMA 12.5.7. Wo. := W 0 (T, V) centralizes V, so w :=;:: w(G, V) > 0 and
Na(Wo):::; M.
PROOF. Suppose A := V9 :::; T with [A, VJ =/= 1. By 12.5.6, s(G, V) = 3, so
that A E A 3 (T, V) by E.3.10. But Aut.M{V3) ~ L 3 (2) is of 2-rank 2, so we conclude
A ~entralizes Vs. Next Tacts on r 1 and I' 4 , and hence acts on V£ := I'1 /\ I'4 =
(1/\2,1/\3,1 /\ 4), with Aut.M(V 3 ) ~ L 3 (2), so the same argument shows A also
centralizes V£. Similarly A acts on %, so that A :::; C := CM 6 (Vi + V£). By12.5.2.4, V6 is the orthogonal module for L6/R6, so that m(C/R6) = 1; again as
A E A3(T, V), we conclude A:::; CLr(%) = R6· By 12.5.5.1, % is a hyperplane
of Cv(r) for each r E Rf, while by 12.5.2.4, V/V6 is a natural module for L6/ R6
isomorphic to R 6. It follows that V = W, where W := (Cv(r) : r E Rf), and no
hyperplane of R 6 lies in A 3 (T, V). We conclude that m(A) > 3, so that A= R6, and
hence W = (Cv(B): m(A/B):::; 3). Now r(G, V) > 3 by 12.5.6, so W:::; Na(A)
by E.3.32. Hence as V = W, we have symmetry between A and V. As A= R6,
V6 = [V,AJ; then by symmetry between A and V, [A, VJ is conjugate to V/ in L^9.
Thus we may take g E G 6 , so g E NM(V 6 ) by 12.5.4, and hence g E Mv by 12.2.6,
contrary to [V, V^9 J =/= 1. This contradiction shows W 0 :::; Cr(V) = 02 (LT), and so
Na(Wo):::; M by E.3.34.2. DLet U := (Va^1 ) and G\ := GifV1.
LEMMA 12.5.8. (1) V :S 02(G1).
(2) U is elementary abelian.PROOF. Let Y := 02 (M1) and U1 := (V?^1 ). By 12.5.2.1, Y has chief series
0 < V1 < V1 < V. Thus Hypothesis G.2.1 is satisfied with Y, G 1 , G 1 , Y, V1 in the
roles of "L, G, H, Li, V", so by G.2.2, U1 E R2(G1) and U1:::; fh(Z(02(G1))). In
particular, Vi :::; 02 ( G1)..
Next V = [V, LiJ :::; [02(L1), Li], so if Ki =Li or Ki/02(K1) ~ SL2(7)/ E4g,
then V:::; 02(K1):::; 02(G1), and hence (1) hold in these cases. We assume that (1)
fails, so Ki := Ki/02(K1) ~ L5(2), M24, or J4 by 12.5.3. Also V ~ V/Vi is the
natural module for Li/0 2 (Li) ~ L 3 (2). Then [V, U 1 J:::; VnU 1 = Vn02(G1) =Vi.
Further V is invariant under Mi by 12.2.6, and from the discussion in the proof of
12.5.3, Mi /02(Mi) ~ L2(2) x L3(2), with the embedding of Mi in Ki determined.
When Ki ~ M24 or L5(2), this is contrary to the action of Y on 02 (Y) as
the tensor product module of rank 6. Finally suppose Ki ~ J 4. Our discussion
of the embedding of Mi showed that Mi < N, with 02 (N) special of order
23 +i^2 and N /02(N) ~ 85 x L 3 (2). It follows that V = Z(0 2 (N)) ::::] N.
Since Li 1:. Cai CV1 ), Ki = (LrKi) 1:. Cai (U1), and in particular V 1:. Cai (U1);
as we saw [V, U1J :::; V1 and Y is irreducible on V 7 , we conclude [U 1 , VJ = V 7.
Therefore N normalizes [U1, VJ = V7. But this is impossible as V 7 is the tensor
product module for Y /02(Y*) ~ L 2 (2) x L 3 (2), and this action does not extend
to N /02(N) ~.85 x L3(2)..
Thus (1) is established. By 12.5.7, V:::; D 1 (Z(W 0 (0 2 (G 1 ), V)))), so (2) follows
from (1). D