884 13. MID-SIZE GROUPS OVER F2we may choose D to permute with Ll· Then [D,02(DT)]::::; 031(Ho) nT::::; Rl, so
Rl is Sylow in RlD.
We argue as in the proof of 13.2.13: Assume that D f:_ M. Then as Rl E
Syl 2 (R 1 D), B := Baum(R 1 ) E Syb(BD) by E.2.3.2. But B = Baum(Q) by
13.2.4.1, so Q E Syl 2 (QD). Then by Theorem 3.1.1 applied with Q, LT, DT
in the roles of "R, M 0 , H", 02 ((LT,DT)) #- 1, so that D ::::; M = !M(LT),
contrary to our assumption that D f:_ M. Therefore D ::::; M = Na(L). Now
as Li/02(L1) is inverted in CT(D), D centralizes L/02(L), so L acts on Y :=
02 (D0 2 (LT)) = (DT), and hence Na(Y)::::; M = !M(L) by Theorem 4.3.2. Then
Lo::::; NH 0 (Y)::::; Hon M, so H::::; Ho= DL 0 T::::; M, for our usual contradiction to
Hf:_M.
This contradiction completes the proof of Theorem 13.2. 7.
13.3. Starting mid-sized groups over F2, and eliminating Us(3)
In this section, with the preliminary results from sections 13.l and 13.2 in hand,
we begin to treat those pairs L, Vin the Fundamental Setup (3.2.1) which constitute
the main topic of the chapter: the pairs such that L / 02 ( L) is an intermediate-sized
group A 5 , A6, A 6 , or U 3 (3) over F 2. As in the previous chapter, we begin by stating
our working hypothesis for this chapter, which exclud.es the groups in the MainTheorem which have arisen in previous sections. In particular, Hypothesis 13.3.1
extends Hypotheses 12.2.3 and 13.1.1. Each section treats one or more pairs L, V
in the FSU; the treatment of a given case assumes the existence of L E Lt(G, T)
with L/CL(V) of the given type.
We also recall, as mentioned in the introduction to the chapter, that to avoid
repetition of arguments, we treat the case L / 02 (L) £::! A5 simultaneously with the
other cases. However in the actual logical sequence, that case is the final one in
our treatment of the FSU, so we actually consider it only when all other groups
have been eliminated. This necessitates the assumption in part (4) of Hypothesis
13.3.1; the effect of this part of Hypothesis 13.3.1 is that we choose LE Lt(G, T)with L/02(L) £::! A5 only when we are forced to do so, because no other choice
is possible. Thus for the purposes of the proof of the Main Theorem, Hypothesis
13.3.1.4 and the results in this chapter which depend on it, are actually invokedonly when we reach that final case.
Thus in section 13.3 and indeed for the remainder of the chapter, we assume
the following hypothesis:HYPOTHESIS 13.3.1. (1) G is a simple QTKE-group, TE Syl 2 (G), and L E
Lt(G,T).
(2) G is not a group of Lie type over F 2 n, with n > 1.
(3) G is not L4(2), L5(2), Ag, M22, M23, M24, He, or J4.
(4) If L/02(L) s::! A5, then K/02(K) s::! A5 for each KE Lt(G, T).The next result describes the members K of£ f ( G, T) which can arise under Hy-
pothesis 13.3.1; as in Remark 12.2.4 of the previous chapter, we can usually replace
our chosen pair L, Vin the FSU by K, VK for some suitable VK E Irr +(K, R 2 (KT)).
LEMMA 13.3.2. If KE Lt(G,T), then
(1) K/02(K) £::! A5, L3(2), A5, A5, or U3(3).
(2) K :::1 KT and KE Lj(G,T). Hence Na(K) = !M(KT).