584 3. DETERMINING THE CASES FOR LE .C.j(G, T)abelian, and thus (5) implies (6) in this case. Now assume the hypothesis of (6b).
We may take g E Na(U) by (3), so
W+, V%):::; 02(Na(U)):::; s n S^9 :::; NM(U) n NM(U^9 ):::; NM(V+) n NM(V%),where the last inclusion holds since V+ is a TI-set under M. Reversing the roles of
V+ and V% if necessary, we may assume that m(V%/Cv+(V+)) 2: m(V+/Cv+(V%)).
Thus as AutL 0 r(V+) is not an FF-module by hypothesis, [V+, V%J = 1. This com-
pletes the proof of (6).
As LoTnormalizes 02(LoT)nCM(V+) = Cr(V+), M = !M(NNM(V+)(Cr(V+))).
Thus Hypothesis E.6.1 is satisfied with V+ in the role of "V", so part (7) follows
from E.6.26 with 1 in the role of "j". DSometimes in arguments where we can pin down the structure of a pair in
the FSU (especially when we can show L is a block), we encounter the following
situation:LEMMA 3.2.11. Assume the Fundamental Setup (3.2.1). Assume further that
V = 02(LoT). Then 02(M) = V = Ca(V) and M = Mv. If further Mv = Lo'l',
then Mv = M = LoT.
PROOF. By A.1.6, 02(M) :S 02(LoT) = V :S 02(£0) :S 02(M), so that
02 (M) = V, and in particular M = Mv as ME M. Now as F*(M) = 02(M),
Ca(V) :::; Z(02(M)) :::; V, so that Ca(V) = V. The result follows. D
Our last two results of the section involve the collection S(G, T) of Definition
1.3.1, and appearing in case (ii) of the hypothesis of 3.2.2.DEFINITION 3.2.12. Define S_(G, T) to consist of those XE S(G, T) such that
either
(a) Xis a {2, 3}-group, or(b) X/02(X) is a 5-group and Auta(X/02(X)) a {2, 5}-group.
Set B+(G,T) := S(G,T)-S_(G,T).
LEMMA 3.2.13. Sj(G, T) s;;; S_(G, T).
PROOF. Assume XE Sj(G, T). Then X/0 2 (X) ~ EP2 or pH^2 for some odd
prime p, and Tis irreducible on X/0 2 ,w(X). By 1.3.7, M = !M(XT), where M :=
Na(X). Let (XT)* := XT/Cxr(R 2 (XT)). By A.4.11, V := [R 2 (XT),X] -j. 1,so as T is irreducible on X/02,w(X), Cx(V) :::; 02,w(X). Thus as R := 02 (XT)
centralizes R2(XT), X = F(XT), so as X is faithful on V, also XT* is
faithful on V. Hence Cr(V) = R and V E R 2 (XT). Therefore the hypotheses
of Theorem 3.1.8 are satisfied with X in the role of "Lo", so ij := q(XT, V) :::;
2 by 3.1.8.1. As ij :::; 2, D.2.13 says p = 3 or 5. We may assume by way of
contradiction that X t/:. S_(G,T), sop= 5 and Auta(X/0 2 (X)) is not a {2,5}-
group. By D.2.17 and D.2.12, X* = Xi x · · · x x; and V = V1 EB··· EB Vs,
where Xi ~ Z5, Vi := [V,Xi] is of rank 4, ands:::; 2. As m 5 (X/0 2 ,w(X)) = 2,
s = 2. As TE Syb(Na(X)), RE Syb(Ca(X/02(X))) by A.4.2.5; so by a Frattini. Argument, Auta(X/02(X)) = AutH(X/02(X)), where H := Na(X) n Na(R).
Thus AutH(X/02(X)) is not a {2, 5}-group, so AutH(X*) is not a {2, 5}-group.
As R centralizes R2(XT), R2(XT) :::; D1(Z(R)). Then as V:::; R 2 (XT),
Cxr(D1(Z(R))) :S Cxr(V) :S R02,w(X),