3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 585so D1(Z(R)) is 2-reduced. Therefore R 2 (XT) = D 1 (Z(R)), so H acts on
[D1(Z(R)),X] = [R2(XT),X] = V,
and hence 02 (H) acts on Vi and Xi. This is a contradiction as AutH(X*) is not
a {2, 5}-group, but Aut(Z5) is a 2-group. DLemma 3.2.13 allows us to establish a result about those LE .C(G, T) such that
L/02(L) is not quasisimple. Recall from chapter 1 that '2p(L) is 02 (Xp) where Xp
is the preimage of D1(0p(L/0 2 (L))).
LEMMA 3.2.14. If LE .C(G, T) and L/02(L) is not quasisimple, then 000 (L)
centralizes R2 (LT).PROOF. We assume L is a counterexample, and it remains to derive a contra-
diction.By 1.2.1.4, L/02,F(L) ~ SL2(q) for some prime q > 3, and T normalizes L by
1.2.1.3. Set V := R 2 (LT); by hypothesis [V, L] # 1 so LE .C1(G, T).
Let L s K E .c*(G, T); then K E .Cj(G, T) by 1.2.9. In the cases in A.3.12where "B/02(B)" is not quasisimple, either 000 (L) S 000 (K) in case (21) or (22),
or K/02(K) ~ (S)L3(r) for some prime r > 3 in case (9). In the latter case by
3.2.3, K is listed in one of 3.2.5, 3.2.8, or 3.2.9, but of course (S)L 3 (r) for a prime
r > 3 does not appear on any of those lists. Thus 000 (L) s 000 (K), so replacing
L by K, we may assume LE .Cj(G,T).
Let 1r := 1r(02,F(L)/02(L)), p E ?r, and X := '2p(L). Since LE .Cj(G,T),X E s;aa(G,T) by the definition in chapter 1, so X E S*(G,T) by 1.3.8. As
AutL(X/0 2 (X)) contains SL 2 (q) for q > 3, X tJ. S_(G,T), so X centralizes V by
3.2.13. HencepEn
Let Ip := OP' (0 00 (L)). If Ip centralizes V for each p E ?r, then 02,F(L) S02(L)Y S CL(V), so 000 (L) centralizes V as L/02,F(L) ~ SL2(q) and V is 2-
reduced. Thus as L is a counterexample, there is p E 1f such that I := Ip does not
centralize V, so I# Xp and hence case (d) of 1.2.l.4 holds and I/0 2 (I) ~ z;e for
some e > l. As case (d) of 1.2.l.4 holds, L/02,F(L) ~ SL2(5). Since e > 1, we
conclude from A.l.30 that p > 5.
Set R := CT(V). As V = R2(LT), 02(LT) s R. As L/02,F(L) ~ SL2(5),
02 (LT) = 02 (IT), and then as [I, V] # 1, R = 02(IT) and V E 1?..2(IT). As
XE 3*(G, T), M := Na(X) = !M(XT) by 1.3.7. As LE C*(G, T), L :':::) M, so
as I char L, I :':::] M. Thus for each HE 1i*(T, M), H n M normalizes I, so case
(I) of Hypothesis 3.1.5 is satisfied with IT in the role of "Mo". As M = !M(XT),02 ( (IT, H)) = 1, so conclusion (2) or (3) of Theorem 3.1.6 holds. In either case
q(IT/CIT, V) s 2. Asp> 5 and [V,J] # 1, this contradicts D.2.13. D
3.3. Normalizers of uniqueness groups contain Na(T)
The bulk of the proof ,of the Main Theorem analyzes the situation whereC1(G, T) is nonempty, leading (as we saw in 3.2.3) to the Fundamental Setup
(3.2.1) and the extended analysis of the cases arising there. The very restricted