0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 491a "faithful action", write Xf for those XE X such that (fh(Z(0 2 (X))),X) # 1,
with a similar use of the subscript to define subsets £, f ( G, T) and 3 f ( G, T). Ouranalysis focuses on the faithful uniqueness groups U in Cj(G,T) and Bj(G,T).
If Y E 7i(T), so that F*(Y) = 02(Y) by 1.1.4.6, then by a standard lemma
B.2.14, V := (ZY) is elementary abelian and 2-reduced: that is, 02 (Y/Cy(V)) = 1.
Following Thompson, define R 2 (Y) to be the set of 2-reduced elementary abeliannormal 2-subgroups of Y. By B.2.12 (26.2 in (GLS96)), the product of members
of R2(Y) is again in R2(Y), so R2(Y) has a unique maximal member R 2 (Y). Weregard R 2 (Y) as an F2Y-module.
Observe that if L E Cj(G, T) with L/0 2 (L) quasisimple, or X E 'Bj(G, T),
then Cu(R2(U)) :::; 02,if!(U).^4 Then the representation of U/Cu(R2(U)) on R 2 (U)
(or indeed on any V E R2(U) with V i. Z(U)) is particularly effective, since for
any weakly closed subgroup W of Cr(V), Wis normal in the uniqueness subgroup
U, so that Nc(W) :::; M. That is M = !M(U) contains the normalizers of various
weakly closed subgroups W of T.
For M := N 0 (0^2 (U)) and U a uniqueness subgroup of the form (L, T) with LE
£*(G,T), or XTwith XE 'B*(G,T), we prove in Theorem 3.3.1 that Na(T):::; M.It follows that T is not normal in H in those cases, so that H is a minimal parabolic
in the sense of Definition B.6.1, and hence we can use the explicit description of
H/0 2 (H) from section E.2 mentioned earlier.We next turn to Theorem 3.1.1, which is used in a variety of ways; it says:
Theorem 3.1.1 If Mo, H E 7i(T), such that Tis in a unique maximal sub-
group of H, and R:::; Twith RE Syb(0^2 (H)R) and R :::;! Mo, then 02((Mo,H)) #
1.For example in our standard setup we can take Mo to be the uniqueness group U
and R := Cr(V)-and conclude that R r:J. Syl2(0^2 (H)R), since Hi. M =!M(U);
hence 02 ( (U, H)) = 1. In particular we use Theorem 3.1.l to rule out the first case
which occurs in Stellmacher's qrc-lemma D.1.5 (see below), and in the remaining
cases the qrc-lemma gives us strong information on a module V for the action ofU. That information is given in terms of small values of certain parameters, which
we now introduce. For X a finite group, let A^2 (X) denote the set of nontrivialelementary abelian 2-subgroups of X. Given a faithful F2X-module V, define
q(X, V) := min{m(V~f;)(A)) : 1 #A E A^2 (X) such that 0 = [V, A, A]}
and the analogous parameter correponding to cubic rather than quadratic action:
A • • m(V/Cv(A)) 2 ·
q(X, V) := mm{ m(A) : 1 # A E A (X) such that 0 = [V, A, A, A]}.For example V is a failure of factorization module (FF-module-see section B.1)
for X precisely when q(X, V) :::; 1.
Using Theorem 3.1.1 and Stellmacher's qrc-Lemma (see Theorem D.1.5), we
obtain:
Theorem 3.1.6 Let T :::; Mo :::; M E M(T) and H E 1i*(T, M) Assume
VE R2(Mo) with Cr(V) = 02(Mo), and H n M normalizes 02 (Mo) or V. Thenone of the following holds:
4Here 02 ,g;(U) denotes the preimage of the Frattini subgroup ~(U/02(U)); elsewhere we use
similar notation such as 02,F(U), 02,E(U), etc.