508 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALSLEMMA 1.2.11. Let H E 1i with T n H =: TH E Syl2(H), and K E C(H).
Assume z E Z := fh(Z(T)) lies in KnTH and:(a) z E [U, 02 (NH(U))] for some elementary abelian 2-subgroup U of H;
(b) CT(02(H)) S TH.
Then either K is quasisimple or H E 7-ie.
PROOF. Assume K is not quasisimple; we must show H E 7-ie. By A.3.3.1,
E(K) = 1, so F*(K) = F(K), and hence CK(F(K)) s F(K). We claim first that
z centralizes O(K): As His an SQTK-group, mr(O(H)) ::::; 2 for all primes r. Thenhypothesis (a) allows us to apply A.l.26.2 to 02 (NH(U)), [U,0^2 (NH(U))] in the
roles of "X, V", to conclude z E [U, 02 (NH(U))]::::; CH(O(H))::::; CH(O(K)). Next
our assumption that z E KnTH gives z E CK(O(K)02(K)) S CK(F(K))::::; F(K),
and hence z E 02(K) ::::; 02(H).
Let Gz := Ca(z) and Hz := CH(z). As z E 02(H), 02 (F*(H)) s 02 (F*(Hz)),
so it suffices to show Hz E 7-ie. As TH is Sylow in Hand TH S Hz, 02(H) S 02(Hz)
by A.1.6. Therefore using (b),
Co 2 ca.)(02(Hz)) S CT(02(H)) S TH n Gz S Hz.
Then as Gz E 7-ie by 1.1.4.3, we get Hz E 7-ie by 1.1.4.4. D1.3. The set 3*(G, T) of solvable uniqueness subgroups of G
As noted in the Introduction to Volume II, it might happen that there are no
nonsolvable locals H E 7-i(T), so that £(G, T) is empty; in this case we will need
to produce some solvable uniqueness groups. Notice also that any L occurring in
cases ( c) or ( d) of 1.2.l.4 involves interesting (and potentially tractable) solvable
subgroups in 02,F(L).
Motivated particularly by the latter example:DEFINITION 1.3.l. Define 3(G, T) to consist of the subgroups X ::::; G such
that:
(1) X = 02 (X) is T-invariant with XT E 1i,
(2) X/0 2 (X) ~ EP2 or p1+^2 for some odd prime p, and
(3) Tis irreducible on the Frattini quotient of X/0 2 (X).
Notice that each XE 3(G, T) is in 7-ie by 1.1.4.6 and 1.1.3.1, so as X = 02 (X)we see 3(G, T) ~ X.
Subsets 3_(G,T) and B+(G,T) of3(G,T) appear in Definition 3.2.12.
We first collect some useful elementary properties of the members of B(G, T):
LEMMA 1.3.2. Let XE S(G,T). Then
(1) X is a {2,p}-group for some odd prime p and X = 02 (X)P for some
PE Sylp(X).
(2) X =(PX)= (P^02 CX)) and 02 (X) = [02(X), P].
(3) T = 02(X)NT(P) and NT(P) is irreducible on P/iJ!(P).
(4) P = [P,i!!(NT(P))]..
(5) If HE 7-i(XT), then X = 02 (0 2 (H)X).
PROOF. Part (1) is immediate from condition (2) in the definition of 3( G, T)
and Sylow's Theorem. As X = 02 (X) in condition (1) of the definition of 3(G, T),
conclusion (1) now implies
X = (Sylp(X)) = (PX)= (P0^2 (X))