938 13. MID-SIZE GROUPS OVER F2As L 1 centralizes K*, L 1 acts on T n K, so R1 E Syb(KL1R1). Thus if
J(R1)::; CR 1 (U), then B = Baum(0 2 (KR 1 )) by B.2.3.5, whereas we saw K does
not normalize B. Hence K+ = [K+, J(R 1 )+]. Also K acts on [CoH(02(L1)),L1] =:
U 1. If [U 1 , K] = 1 then K ::; Ca(V3) ::; Mv using 13.5.4.4, again contrary to
Ki M. Thus [U1, K] =f. 1, so as [U1, K] ::; Uo, U2 := [Uo, L1, K] =f. 1. Thus we may
take U ::; U 2 , so U = [U, L 1 ]. Then as L 1 centralizes K*, Lt :S! x+, so that Lt ~ Z3
as 02 (X+) = 1. Then since K+ = [K+, J(R 1 )+], it follows from Theorem B.5.6that U is the sum of two isomorphic natural modules for K+ ~ L 3 (2). Therefore
by B.2.20, B+ = J(R 1 )+ is the unipotent radical of a minimal parabolic K;i Rt
of K+ Rt. Then by B.2.3.4, B = Baum(02(KoR1)), so that Ko ::; Na(B) ::; M.
Then 02 (K 0 )::; 02 (H n M) = L 1 by 13.7.3. But /L1/ 3 = 3 since we are assuming
that L/0 2 (L) is A 6 rather than A 6 , so 02 (Ko) = L 1 , contradicting L1 i K. Thusthe proof of (1) and hence of the lemma is at last complete. D
LEMMA 13.7.5. Let X := L 1 if L/02(L) ~ A5, and X := L1,+ if L/02(L) ~
A 6. Assume KE C(H) and X::; K. Then(1) K :S! H.
(2) Us= [Us,K].
(3) Ns(l/3) ::; M.
(4) If m3(NK(V3)) = 2, then L/02(L) ~ A5 and L1 = B(NK(l/3)).
(5) If L1::; K and m3(NK(V3)) = 1, then L/02(L) ~ A5.
(6) 0
31
(NK(l/3)) is solvable.
(7) If L1 "f:. K, then AutL 1 (K/02(K)) =f. Autx(K/02(K)).PROOF. Since T normalizes X::; K, K :S! H by 1.2.1.3, proving (1). Further
V3 = [l/3,X]::; [Us,KJ, so that Us= (V;,,B) = [Us,KJ, establishing (2). Part (3)
follows from 13.5.5. Then by (3) and 13.7.3, either B(NK(V3)) = L1 or L/0 2 (L) ~
A5 with B(NK(V3)) = X. Now m3(X) = 1, while m3(L1) = 1 when L/02(L) ~ A5,and m 3 (L 1 ) = 2 when L/02(L) ~ A 6 ; so it follows that (4) and (5) hold. As
B(NK(V3)) ::; L 1 which is solvable, (6) holds.
Finally suppose that L 1 i K, but the conclusion of (7) fails. Since X ::;
K, X < L1, so L/02(L) ~ A5; then as (7) fails, L1 = XLe, where Le =02 (CL 1 (K/02(K)). As X and Lo are the only proper nontrivial T-invariant sub-
groups Y of L1 with Y = 02 (Y), it follows that Le= L 0. But then K normalizes
02 (02(K)L 0 ) =Lo and so lies in M by 13.2.2.9, contrary to 13.3.9. D
The next result eliminates various possibilities for H* and its action on Us.
As usual Theorem C (A.2.3) determines the possibilities for n in (1) and (3). Thelemma considers all cases where [Us, K] E Irr+ (Us, K) is an FF-module, except
the cases where the noncentral chief factor for K on Us is the natural module for
K/02(K) ~ L2(2n) or A5.
LEMMA 13.7.6. Assume KE C(H) and let UK:= [Us,K]. Then
(1) If K/02(K) ~ Ln(2) and UK/CuK(K) is the natural K/02(K)-module,then n = 4, Us is the natural module for H* ~ L4(2), and L/02(L) ~ A5.
(2) If K/02(K) ~ L5(2), then UK/CuK(K) is not a 10-dimensional irreducible
for K/02(K).
(3) If K/02(K) ~An and UK/CuK(K) is the natural module, then Us= UK,
L1::; K, H =KT, and applying the notation of section B.3 to UH, either