1.3. THE SET B*(G, T) OF SOLVABLE UNIQUENESS SUBGROUPS OF G 509and 02(X) = [02(X), P], giving conclusion (2). Notice XT =PT, so the Dedekind
Modular Law gives Nxr(P) = PNr(P); then a Frattini Argument onX = 02 (X)P
gives T = 02(X)Nr(P). Now Nr(P) is irreducible on P/i!!(P) by condition (3) ofthe definition of S(G, T), so conclusion (3) is proved.
Let S := Nr(P) and S := S/Cr(P). Now S is irreducible on P/iJ!(P) by (3),
so each involution i E Z(S) inverts P/iJ!(P). Thus for each I~ S with i E J,
P = [P, I]. In particular if i!!(S) -=f. 1, we can choose I= iJ!(S), so that (4) holds
in this case. Otherwise i!!(S) = 1, and then S* is reducible on P/i!!(P) by A.1.5.
This contradiction completes the proof of ( 4).
Under the hypotheses of (5), 02(H) ~ T, while by condition (1) of the defini-
tion, T ~ Na(X) and X = 02 (X), so X = 02 (0 2 (H)X), as required. D
Assume for the moment that LE .C(G, T) with L/0 2 (L) not quasisimple, as in
cases (c) and (d) of 1.2.1.4. Then Lis T-invariant by 1.2.1.3. Given an odd prime
p, define
Sp(L) := 02 (Xp), where Xp/02(L) := f21(0p(L/02(L)));
then define Sraa(G,T) to be the collection of subgroups Sp(L), for LE .C(G,T)
with L/0 2 (L) not quasisimple, andp E n:(F(L/0 2 (L))).
We observe that XE Sraa(G, T) satisfies conditions (2) and (3) in the definition
of S( G, T), using the action of L/0 2 ,p(L) ~ SL 2 (r) (r = p or 5) in cases (c) and (d)
of 1.2.1.4. By construction, X = 02 (X), while Xis T-invariant as X char L ::::l LT.
Finally LT E 1-ie by 1.1.4.6, so that 1 -=/:- 02 (LT) ~ 02 (XT) by A.1.6, the last
requirement of condition (1) of the definition. So we see:
LEMMA 1.3.3. Sraa(G, T) ~ S(G, T).
Define S* ( G, T) to consist of those X E S ( G, T) such that XT is not contained
in (L, T) for any L E .C( G, T) with L / 02 ( L) quasisimple. So for S (in contrast
to .C), the superscript * will not denote maximality under inclusion in the poset
'B(G, T). However the following result will be used in 1.3.7 (which is the analogue of
1.2.7.1) to prove that XT is a uniqueness subgroup for each member X of S*(G, T).
Furthermore the list of possible embeddings of members of 'B(G, T) in nonsolvable
groups appearing in the lemma will also be very useful.
PROPOSITION 1.3.4. Let X E S(G, T), P E Sylp(X) a complement to 02 (X)
in X, and HE 1-i(XT). Then either X ::::l H, or X ~ (LT) for some LE C(H)
with L / 02 (L) quasisimple, and in the latter case one of the following holds:
(1) Lis not T-invariant and P = (P n L) x (P n L)t ~ Ep2 fort E Nr(P) -
Nr(L). Either L/0 2 (L) ~ L 2 (2n) with n even and 2n = 1 mod p, or L/0 2 (L) ~
L2(q) for some odd prime q.
In the remaining cases, L is T-invariant and satisfies one of:
(2) P ~ Ep2 and L/02(L) ~ (S)L3(p).
(3) P ~ Ep2, L/02(L) ~ Sp 4 (2n) with n even and 2n
Autr(P) is cyclic.
(4) p = 3, P ~ Eg, and L/02(L) ~Mn, L4(2), or L5(2).
1 mod p, andPROOF. Set H := H/0 2 (H). We first consider F(H). So let r be an odd prime,
and Ra supercritical subgroup of Or(H). (Cf. A.1.21). As usual mr(R) ~ 2 since
mr(H) ::::; 2. Therefore by A.1.32, [R, P] = 1 if p -=f. r; while if p = r, then either