892 13. MID-SIZE GROUPS OVER F2If 02 (L+) = 1, then 02(L+) s Q 1 s Ca(Vs) by (a), impossible as L+ induces
A4 on Vs/V1.(d) 02 (L+) centralizes F(Gi).
Assume 02 (L+) is nontrivial on Op(Gi) for some odd prime p. Then as
L+/0 2 (L+) has order 3, Auto 2 (L+)(Op(Gi)) is noncyclic, so by A.1.21 and A.1.25,there is a noncyclic supercritical subgroup P of Op(Gi) such that P ~ Ep2 or p1+^2
and Aut(P)/Op(Aut(P)) is a subgroup of GL 2 (p). Hence AutL+(P*) ~ SL2(3).
Let P := 02 (P+), where P+ is the preimage of P* in G1. Then PL+T =: H E
1i(T) n G 1. Further as L+ is normal in M 1 but L+ is not P-invariant, P 1:. M.
As AutL+(P*) ~ SL 2 (3), PE B(G,T) with AutrnL+(P*) ~ Qs. Also [U,P] =f=.
1 as mp(Y) = 1 by (b). Since U s fh(Z(Q1)) by (a), P E 'Bt(G,T) by anapplication of A.4.9 to P, G 1 in the roles of "X, M". Assume P S (KT) for
some K E C(G, T) with K/0 2 (K) quasisimple. Then (KT) is described in 1.3.4.
Further K E .Ct(G, T) by 1.3.9.2, so K = (KT) by 13.3.2.2, and K/02(K) isdescribed in 13.3.2.1. As the lists in 1.3.4 and 13.3.2.1 do not intersect, there is no
such K, so PE Bj(G, T). Then by 3.2.13, PE S_(G, T). Since Auta(P/02(P))
involves SL 2 (3) which is not a {2, 5}-group, we conclude from Definition 3.2.12
that Pis a {2,3}-group, so that p = 3. As m3(PL+) s 2 with AutL+(P*) ~
SL 2 (3), we conclude P/0 2 (P) ~ P* ~ E 9 rather than 31+^2. Let W := R 2 (PT);
as AutT(P*) ~ Qs, we conclude from D.2.17 that q(AutpT(W), W) > 2. HoweverNa(P) = !M(PT) by Theorem 1.3.7, so that we may apply Theorem 3.1.8.1 to P,
Win the roles of "Lo, V" to obtain q(AutpT(W), W) s 2, contrary to the previous
observation. This contradiction completes the proof of (d).
Since 02(Gi) = 1, (c) and (d) say there is KE C(G1) with K* a component ofG]' and [K*, 02 (L+)J =f=. 1. By 1.2.1.3, L+ = 02 (L+) normalizes K. In particular,
K/02(K) is quasisimple and K = [K,L+l·
(e) KE .Cj(G, T), G 1 s Na(K) = !M(KT), L 1:. Na(K), and K 1:. M.First [U, K] =f. 1 by (b), so using (a) and A.4.9 as in the proof of (d), K E
.Ct(G,T). Then by 13.3.2.2, K E .Cj(G,T), K ::::! KT, and G 1 s Na(K) =
!M(KT). As G1 1:. M, · Na(K) =f. M. So as M = !M(LT), L 1:. Na(K), and as
Na(K) = !M(KT), K 1:. M, completing the proof of (e).
If L2 s Na(K), then L = (L1,L2) s Na(K), contrary to (e). So:
(f) L2 f:. Na(K).
As L2 1:. Na(K) ;:::: Ca(Z) by (e) and (f), L 2 T contains some HE 1i*(T, Na(K)),
and H 1:. Ca(Z). By (e), KE .Cj(G,T), and we saw K/0 2 (K) is quasisimple, so02(KT) = CT(R2(KT)) by l.4.l.4b. Then applying 3.1.8.3 to K, R2(KT) in the
roles of "L, V", K = [K, J(T)]. Set J := KL+T, W := R 2 (J), J+ := J/CJ(W),
and WK:= [W,K]. Then R2(KT) s W by A.1.11, so that Irr+(K,R 2 (KT)) ~
Irr +(K, WK). Now K/0 2 (K) is described in 13.3.2.1, and the members of the set
Irr +(KT, R2(KT), T) are described in 13.3.2.3. Thus applying Theorem B.5.6 tothe FF-module WK for J, we conclude that either WK E Irr+(KT,R 2 (KT),T) or
K/02(K) ~ Lg(2) and WK is the sum of two isomorphic natural modules.
(g) L+ 1:. K.
Assume that that L+ SK. Define V(K) as in Definition A.4.7, and set J :=
J/CJ(V(K)). By (a) and A.4.8.4, Vs= [Vi,L+] s [01(Z(Qi)),K] s V(K). Let X
be of order 3 in L+ and set QJ := 02(J). By A.4.8.1, QJ centralizes X. Thus by