12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 797LEMMA 12.2.6. V is a TI-set in M, so if 1 # U ::=:; V and H ::=:; Na(U), then
HnM=NH(v):LEMMA 12.2.7. Assume Ca(Z) ::=:; M and H E 1i*(T, M). Let K := 02 (H)
and VH := (ZH). Then
(1) VH E R2(H) and CT(V) = 02(H) ::=:; CH(VH) ::=:; kerNH(V)(H).
Assume further that H is not solvable. Then(2) K/02(K) ~ L2(4).
(3) K ::=:;YE Cj(G,T), and either
(i)Y =Kand [VH,K] is the sum of atmosttwoAs-modulesfor K/0 2 (K),
or(ii) Y/02,z(Y) ~ A1, Hypothesis 12.2.3 is satisfied with Y in the role of
"L ", and for each Vy E Irr+ (Y, R 2 (YT), T), Vy is T-invariant and m(Vy) = 4 or
6.
PROOF. First VH E R2(H) by B.2.14, so 02 (H) ::=:; CH(VH)· Then as Ca(Z) ::=:;
M but H 1:. M, K 1:. CH(VH)· The remaining statements in (1) follow from B.6.8.6
and 12.2.6.
Now assume H is not solvable. By E.2.2, K = (XT) for a suitable X E C(H)
with X/02(X) quasisimple and X 1:. M. As [VH,X] # 1 by (1), XE C1(G,T).
We may embed X ::=:;YE C*(G,T), and then by 1.2.9, YE Cj(G,T).
Suppose first that Y/0 2 (Y) is quasisimple. Then by Remark 12.2.4, Hypothesis12.2.3 is satisfied with Y in the role of "L". In particular Y is T-invariant; and
for each Vy E Irr +(Y, R2(YT), T), Vy is T-invariant, and Y, Vy satisfies one of the
conclusions of 12.2.2.3. Therefore Tacts on X by 1.2.8.1, so that X = K with KT
described in E.2.2.2.Assume first that K = Y. Comparing the lists of 12.2.2.3 and E.2.2.2, we
conclude that K/02,z(K) ~ L2(4), L3(2), or A5. However if K/02,z(K) is L3(2)
or A5, then by E.2.2.2, T is nontrivial on the Dynkin diagram of K/0 2 (K), a
contradiction as 12.2.2.3 says Vy/Cvy(K) is a natural module. Thus K/0 2 (K) ~
L 2 (4). Hence by the exclusions in Hypothesis 12.2.3 and Theorem 6.2.20, [VH,K]is the sum of at most two As-modules. Therefore (2) and (3i) hold in this case.
So we may assume that K < Y. Therefore by 1.2.4, the embedding of Kin
Y is described in A.3.12. Searching for pairs K, Y with K appearing in E.2.2.2
and Y appearing in 12.2.2.3, we conclude that either K/02,z(K) ~ L2(4), L3(2),
or A5, with Y/02,z(Y) ~ A1; or K/02(K) ~ L3(2), with Y/02(Y) ~ L4(2) or
Ls(2). But again when K/0 2 (K) is L3(2) or A5, T is nontrivial on the Dynkin
diagram of K/02(K), whereas there is no such embedding of KT/02(KT) in 87
or Aut(L4(2)), so KT/02(KT) ~ Aut(L3(2)) and YT/02(YT) ~ Aut(Ls(2)).
However this is also impossible as Vy is a T-invariant natural module for Y/0 2 (Y)
by 12.2.2.3, so Y/0 2 (Y) is self-normalizing in GL(Vy). Thus K/02(K) ~As and
Y/02,z(Y) ~ A1, so that (2) holds. Also as Y, Vy appears in 12.2.2.3, (3ii) holds.
Finally assume that Y/0 2 (Y) is not quasisimple. Then by 1.2.1.4, Y/0 2 , 2 1(Y) ~
SL 2 (p) for some prime p > 3. Now Tacts on Y by 1.2.1.3, so again X = K is T-
invariant by 1.2.8.1, and hence appears in E.2.2.2; in particular as K/0 2 (K) is qua-
sisimple, K < Y. Again by 1.2.4, the embedding X < Y is described in A.3.12, so
by A.3.12, K/02(K) ~ L2(P) or L2(5). Now for some odd prime q, Xo := Sq(Y) E
Brad(G, T), and as Y E C(G, T), by definition Xo E s;ad(G, T). Then by 1.3.8,
Xo E S(G,T). By 3.2.14 applied to Y, [Z,Xo] = 1, so XoT::::; Ca(Z)::::; M. Then