504 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS
(3) If L E C(H), then either L :::] H; or [LH[ = 2 and L/02(L) ~ L2(2n),
Sz(2n), L 2 (p), pan odd prime, or J 1.
(4) Let LE C(H) and H = H/0 2 (H). Then one of the following holds:
(a) L is a simple component of H on the list of Theorem C (A.2.3}.
(b) L is a quasisimple component of H, Z(L) ~ Z3, and L is SL3(q),
q = 2e or q an odd prime, A5, A7, or M22·
(c) F(L) ~ Ep2 for some prime p > 3, and F(L) affords the natural
module for L/F*(L) ~ SL2(p).
(d} F(L) is nilpotent with Z(L) = <J?(F(L)), L/F*(L) ~ SL2(5), and
for each p E n(F*(L)):
(i} either p^2 = 1 mod 5 or p = 5; and
(ii} either Op(L) ~ p1+^2 , or Op(L) is homocyclic of rank 2.
(5) If LE C(H) satisfies 02 (L) ::::; Z(L) and m2(L) > 1, then L is quasisimple.
PROOF. As we observed at the start of the section, since H E 1-i, H is an
SQTK-group, and hence so is H := H/0 2 (H). Certainly 02 (H) = 1-so we may
apply the results of section A.3. to H. Further by A.3.3.4:
(*) The map L f--7 L is an H-equivariant bijection of C(H) with C(H)-with
inverse R f--7 K^00 , where K is the full preimage of R in H.
Thus for LE C(H), we have LE C(H) and the possibilities in (4) are just those
from A.3.6. Similarly the existence of the equivariant bijection in (*),together with
A.3.7, A.3.9, and (1) and (3) of A.3.8, implies (2), (1), and (3), respectively.
Assume the hypotheses of (5). If L/0 2 (L) is quasisimple, then as 02 (L) ::::; Z(L)
and Lis perfect, Lis quasisimple. Thus we may assume that case (4c) or (4d) holds.
Then as 02 (L) ::::; Z(L), 02 ,p(L) = 02 (L) x O(L). Thus L/O(L) is the central
extension of the 2-group 02 ,p(L)/O(L) by L/0 2 ,p(L) ~ SL 2 (p). But the multiplier.
of SL2(P) is trivial (I.1.3), so we conclude 02 (L) = 1. Now m2(L) = m2(L/O(L))
and L/O(L) ~ SL 2 (p) has 2-rank 1, contrary to the hypothesis that m2(L) > 1.
This establishes (5), and completes the proof of 1.2.1. D
As we mentioned in the Introduction to Volume II, in the bulk of the proof,there will be H E 1i with H nonsolvable; and in that case by 1.2.1.1, C(H) is
nonempty. ·
LEMMA 1.2.2. Let HE 1-i, H := H/02(H), LE C(H), and p an odd prime.
(a) If [LH[ = 2 and p E n(L), then OP' (H) = (LH).
(b} If mp(L) = 2 then L :::! H.
PROOF. Part (b) follows as mp(L) = 1 for each of the groups L listed in 1.2.1.3.
Assume the hypotheses of (a), and set Lo := (LH). Recall Lo is normal in
H by 1.2.1.3. Then mp(Lo) = 2, so CFI(Lo) is a p'-group as mp(H) ::::; 2. As
[LH[ = 2, OP' (H) normalizes L. Recall from the Introduction to Volume I that we
refer to [GLS98] for the structure of the outer automorphism groups of the groups
listed in Theorem C. For those L listed in 1.2.1.3, 02 (0ut(L)) is a group of field
automorphisms (or trivial), and 02 (Aut(L)) splits over Inn(L) ~ L. Therefore if
OP' ( H) 1:. Lo, there is x of order p in NH ( L) - Lo. Then x centralizes nontrivial
elements of order pin each factor of PE Sylp(L 0 ), contradicting mp(H)::::; 2. This
contradiction gives OP' (H) ::::; L 0 , while Lo= OP' (Lo) as Lis simple and p E n(L).
This proves (a). D