. 666 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL
(b) Ki/0 2 (Ki) s=: L~(p), and there is an X-invariant K2 E .C(G, T) n Ki with
K202(Ki)/02(Ki) ~ 8L2(p).
In case (b), if the projection of X 5 on Ki centralizes K2/02(K2), then from
the structure of L~(p), X 5 centralizes a Sylow 2-group of Ki/02(Ki), which is not
the case as X does not normalize TnK 0. Thus the projection is inverted in TnK2,
so as X :::;J XT, X s K 2. Similarly in case (a) the projection is inverted in TnKi,
so X s Ki. Now L n Mi contains TL and so is contained in a Borel subgroup of
L, and hence X :::;J Min M. Thus in case (b), K 2 i M as X < K2. In this
case, we replace Ki by K 2 , reducing to the case where Ki E .C(G, T), Ki i M,
and Ki/0 2 (Ki) s=: L 2 (pe) as in case (a). (We no longer require Ki E C(Mi)). As
X s Ki is normalized by T, Ko = (K[) =Ki.Let K{T* := KiT/02(KiT). Recall by Remark 6.1.2 that M = !M(Nc(J(T))
and J(T) = J(0 2 (LT)) :::;J XT. Thus J(T) is not normal in KiT as Ki i M, so
there is A E A(T) with A* -/= 1. As J(T) ::::1 XT, A* s J(T)* s 02(X*T*). Butfrom the structure of Aut(L 2 (pe)), each nontrivial elementary abelian 2-subgroup
of 02(X*T*) is fused under Ki to a subgroup of T* not in 02(X*T*), contrary to
J(T)* s 02 (X*T*). This contradiction finally completes the proof of 6.1.4. DLEMMA 6.1.5. 'H*(T, M) ~ Ca(Z).
PROOF. Assume that H E 'H*(T, M) with [H, ZJ -/= 1, and let K := 02 (H).
Let DL be a Hall 21 -subgroup of NL(TL)· Enlarging V if necessary, we may take
V = R2(LT), so Zs V. By 5.1.7.2, K = [K, J(T)] and L = [L, J(T)].
Let VL := VL/CvL(L) and ZL :=Zn VL. As VL is the natural module for L
by 6.1.4, J(T) = rh by B.4.2.1. Hence J(T) S TLQ where Q := 02(LT), so DL
normalizes J(TLQ) = J(T). Also VL = [ZL,L] and CLT(VL) = CLT(VL) = Q.Let 8 := Baum(T). As L = [L, J(T)], and VL is the natural module, E.2.3.2 says
8 E 8yl2(L8) and hence 8nL E 8yl2(L). As J(T) =TL and TLQ = CT(Cv(TLQ)),
also 8 = Baum(TLQ), so that DL normalizes 8.
As VL is the natural module for L, the normalizer N of L ~ 8L 2 (2n) in GL(VL)
is I'L2(2n), with CN(L) ~ Z 2 n-i, and 02 (CN(ZL)) is the product of T with a
diagonal subgroup of CN(L) x L isomorphic to Z2n-i. Therefore Cz := CM(ZL)
acts on TL and on [ZL, L] = VL, and 02 (Gz/TL) is a subgroup of Z 2 n_i.
Let UH:= (ZH) and set fI := H/CH(Ujf). Observe UH E R 2 (H) by B.2.14.By Hypothesis 6.1.1, n(H) = 1. Recall by 3.3.2.4 that we may apply results of
section B.6 to H. So as K = [K, J(T)] and [H, Z] -/= 1, H appears in case (2)
of E.2.3, with fI ~ 83 or 83 wr Z 2 and 8 E 8yl 2 (K8). By parts (a) and (b) of
B.6.8.6, CT(UH) ~ H.
We claim CH(UH) = 02(H), so assume otherwise. By B.6.8.6.a, CH(UH) s02,if!(H), so by B.6.8.2, H/02(H) ~ Ds/31+^2. Thus there is a T-invariant subgroup
Y = 02 (Y) of 02,if!(K) with Y = [J(T), Y] and IY: 02 (Y)I = 3, and Y centralizes
UH by assumption. Then by B.6.8.2, Y S 02 ,if!(K) SM, so as Y centralizes UH
and Z s UH, Y centralizes Z and normalizes [Z, L] = VL. If Y centralizes VL then
[Y, L] s CL(VL) = 02(L), so that LT normalizes 02 (Y02(L)) = Y, and hence
Na(Y) SM= !M(LT). As KS Na(Y), this contradicts Ki M. Hence Y-/= 1,
and as Y S Cz, we conclude from paragraph three that J(T) =TL :::;I TY. This
contradicts Y = [Y, J(T)], and so completes the proof that CH(UH) = 02 (H). Itfollows that H = J(H)T with H/02(H) ~ 83 or 83 wr Z 2 , and in particular that
HnM=T.