840 12. LARGER GROUPS OVER Fz IN .Cj(G, T)
F-hyperplane of F V. Hence as Ri, i = 1, 2, are representatives for the conjugacy
classes of 4-subgroups of L, we may take S = R 2 since R 1 centralizes no hyperplane
by 12.7.2.2. Then as [V, R 2 ] is not a point by 12.7.2.4, V does not split over Vt as
an NL(vt)-module. This completes the proof of (2), and (2) and 12.7.2.6 imply (3).
Further 112/vt is the 1-dimensional F-subspace centralized by S = R2, so vt ::; V2
and P 2 has two orbits on F-points of V2 of length 2 and 3, and then (4) follows as
Tacts on V1. D
For the rest of the section, t and vt have the meaning given in 12.7.4. Observe
that Lt/02(Lt) ~ A5 using 12.7.4.2, so that Lt= Lf.
LEMMA 12.7.5. Either{1} Gt ::; M, or
{2} Kt := (Lf•) is a component of Gt with Vt= Z(Kt) and Kt/Vt~ L3(4).
PROOF. Assume that (1) fails, and choose t so that Tt = CT(t) E Syb(Mt)·
From 12.7.4, 02 (Lt'i't) = 1 and V = [V,Lt], and we saw Lt= Lf, so we may apply
12.2.12.2 to conclude that Hypothesis C.2.8 is satisfied with Gt, Mt, Lt, Q in the
roles of "H, MH, LH, R". By C.2.10.1, O(Gt) = 1. By Theorem C.4.8, Lt::; KE
C(Gt) with K/0 2 (K) quasisimple and K described in one of the conclusions of that
result. If conclusion (10) of C.4.8 holds, then for g E Gt - Mt, Lt -=/= Lf ::; Mt, so
Lf ::; B(Mt) =Lt by 12.7.4.2, a contradiction. Thus by C.4.8, Lt < K, K/0 2 (K)
is quasisimple, and K is described in C.3.1 or C.4.1. By 12.7.4.3, Lt = B(Mt) and
V = [V, Lt], so Lt= B(K n M) and t E vt::; V::; Lt::; K, sot E Z(K).
Suppose first that F*(K) = 02 (K). Examining the list of C.4.1 for "Mo" with
Mo/02(Mo) ~ L 2 (4) acting naturally on V/vt, we see conclusion (2) of C.4.1 holds:
K is an Sp 4 (4)-block with vt ::; Z(K), and Mn K is the parabolic stabilizing the
2-dimensional F-space V /Vt in U ( K) /Vt. As U ( K) is a quotient of the orthogonal
FK-module of dimension 5, V splits over vt as an Lt-module--contrary to 12.7.4.2.
Thus as K/0 2 (K) is quasisimple, K is a component of Gt; and Z(K) is a 2-
group since O(Gt) = 1. This time examining the list of C.3.1 for "Mo" given by Ltacting as L2(4) on V/vt, we see that one of cases (1), (3), or (4) must occur. Then
as vt ::; Z(Lt) and t E vt n Z(K), we conclude that vt ::; Z(K). Now by I.1.3, the
only case with a multiplier of 2-rank 2 is K/Z(K) ~ L 3 (4). As Vis Lrinvariant
and elementary abelian, Vt = 02 ( K) from the structure of the covering group of
L 3 (4) in I.2.2.3b. Thus as Z(K) is a 2-group, (2) holds in this case, completing the
proof of 12. 7.5. D
LEMMA 12.7.6. Assume that L is a A 6 -block. Then
(1) Q = 02(LT) = v x CT(L).
{2} If CT(L) = 1 then 02(L) = V = 02(M) = Ca(V) and M =LT.PROOF. Since the 1-cohomology of Vis trivial by l.1.6, (1) follows from C.1.13.b.
Assume CT(L) = 1. By (1), 02 (LT) = V. Now (2) follows from 3.2.11. D12. 7.2. The identification of He. In this subsection we prove:
THEOREM 12.7.7. If Gt i. M, then G is isomorphic to He.
PROOF. Assume Gt i. Mand let Kt := (Lf•). By 12.7.5, Kt is quasisimple