512 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS
We have the following corollaries to Proposition 1.3.4:PROPOSITION 1.3.5. If XE S*(G, T) and HE H(XT), then X :Si H.
PROOF. Notice that the proposition follows from 1.3.4, since by 1.2.6, C(H) ~
C(G, T) for HE H(T). D
LEMMA 1.3.6. If XE S(G,T) with X/02(X) ~p^1 +2, then XE S*(G,T).
PROOF. This is immediate from 1.3.4, which says that if X. :::; (LT) with
L/0 2 (L) quasisimple, then P ~ EP2. D
Now, as promised, we see that if X E 3* ( G, T), then XT is a uniqueness
subgroup of G:
THEOREM 1.3.7 (Solvable Uniqueness Groups). If XE S*(G, T) then Na(X) =
!M(XT).
PROOF. Let M E M(XT). By 1.3.5, X :SI M, so maximality of M gives
M=Na(X). DRecall that Srad(G,T) consists of the subgroups Sp(L), for LE .C(G,T) such
that L/02(L) is not quasisimple; and by 1.3.3, Srad(G,T) ~ S(G,T). Define
s;ad(G, T) to consist of those X E Srad(G, T) such that X :SI LE .c*(G, T). We
see next that XT is a uniqueness subgroup for each XE s;ad(G, T). This fact will
allow us to avoid most of the difficulties caused by those L E .C*(G, T) for which
L/02(L) is not quasisimple, by replacing the uniqueness group LT with the smaller
uniqueness subgroup Sp(L)T.PROPOSITION 1.3.8. s;ad(G,T) ~ 3*(G,T).
PROOF. Let XE s;ad(G, T). Then X :SI LE .C*(G, T) by definition. By 1.3.3,
XE S(G, T), so there is an odd prime p such that X = 02 (X)P for PE Sylp(G).
Indeed by 1.2.1.4, p > 3 and eith~r L/X ~ SL 2 (p) or L/0 2 ,p(L) ~ SL 2 (5). Thus
in any case there is a prime r with L/0 2 ,p(L) ~ SL 2 (r); r has this meaningthroughout the remainder of the proof of the proposition..
By 1.2.1.3, T normalizes L. Then by 1.2.7.3, M := Na(L) = !M(LT). We will
see shortly how this uniqueness property can be exploited. As X is characteristic
in L, X :SI M, so we also get M = Na(X) using the maximality of M EM.
We will next establish a condition used to apply the methods of pushing up.Set R := 02(XT). Recall the definition of O(G,R) from Definition C.1.5. We
claim that
O(G,R):::; M. (*)
The proof of the claim will require a number of reductions.We begin by introducing a useful subgroup Y of Na(R): Recall X :SJ LT, and
R is Sylow in OLr(X/02(X)) by A.4.2.7; so by a Frattini Argument,
LT= OLr(X/02(X))NLr(R). (**)
Thus if we set Y := NL(R)^00 , then Y contains X, and also by the factorization
(**), Ny(P) has a section SL2(r), where r = p or 5. So our construction gives
1 #YE C(NLr(R)), such that Y/02(Y) is not quasisimple. Further T normalizes
R, so in fact using 1.2.6, YE C(NLr(R)) ~ .C(G, T).