804 12. LARGER GROUPS OVER F2 IN .Cj(G, T)Furthemore m(W) :::; 3m(U n V) = 6, so m(W) = 6, since this is the minimal
dimension of a faithful module for P ~ 31+^2.
Let QL := 02 (L) and TL := T n L. Then T = QTL by 12.2.20, while Q =
CLT(V) and LT ~ L 3 (2). As ~ is inverted in TL and <I> permutes with T, <I> is
inverted in NTL (X) so <I> :::; L.
Now K = PW by 12.2.22, and W = [W, <I>], so Y := <I>W = 02 (<I>T). As
<I>W char PW ::;J H, Y ::;J H. Now as L ~ L3(2), 02(Y) = W ~ W/(W n Q)
is of rank 2, as is W n V; so as W is of rank 6, (W n Q)/(W n V) ~ (W n
Q)V/V is of rank 2. Further (W n Q)V/V = [Q/V, <I>] since W = [W, <I>] = 02(Y).
Therefore L has a unique noncentral chief factor [Q, L]/Qo (for some suitable Qocontaining V) on [Q, L]/V. Also since the unique noncentral chief factors for on
[Q, L]/Q 0 and V are in the centralizer of the unipotent radical W, it follows from the
representation theory of L 3 (2) that [Q,L]/Q 0 is isomorphic to Vas an £-module.
Further W[Q, L]/[Q, L] ~ E 4 , so L/[Q, L] is not SL 2 (7); and hence as L = 02 (L),
[Q, L] = QL: As W is abelian by 12.2.22, W centralizes (W n Q)V/V = [Q/V, X],
so QL/V is not the 4-dimensional indecomposable of B.4.8.2. Thus V = Qo. Then
as V:::; Z(Q), while Li~ transitive on (QL/V)#, and W n QL contains involutions
not in V, it follows that QL ~ E54.
Let QH := 02(H). As His irreducible on W, W:::; Z(QH), so CqH(Y) =
CqH(<I>). Each involution in CT(<I>) is in QHV, so from the action of Lon QL,CqL () = (q, v) with q E CqH () = CqH (Y) and QL = (qT) V. We saw Y ::;l H,
so CqH(Y) ::;l H, and hence (qT) :::; CT(Y), contradicting QL = (qT)V. This
contradiction completes the proof of 12.2.23. DBy 12.2.22 and 12.2.23,
K = PW = 02 (I), and U = W <l H.
In particular as P i M by construction,
Ii_M.As V ::;l T, also
I=PWV=KV ::;J KT=H.
Furthermore as Tacts on U = 02 (J) and Q = CT(V),
[Q, U]:::; Cu(V) =Un V:::; V;
and hence as L = (UL), we have:LEMMA 12.2.24. L is ·an Ln(2)-block.We remark that 12.2.24 establishes the first statement in each of conclusions
(2)-( 4) of Theorem 12.2.13, so it only remains to identify G. Our next result showsthat M = L, and hence determines the structure of Ca(z) as Ca(z) :::; M.
By 12.2.20, Un Vis a hyperplane of V, and U induces on V the full group of
transvections with axis UnV on V. Let Y := 02 (NL(UnV)), so that Y/0 2 (Y) ~
Ln-1(2). ThenU=02(YT)=CT(UnV),sothatCT(UnV)=UCT(V) ::;l YT.LEMMA 12.2.25. (1) M = L and V = 02 (L).
(2) n = 3 or4, p = 3, U = Ca(U), Na(U) = YIT, YIT/U ~ Ln-1(2) xL2(2),