500 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS
We begin by defining some notation.DEFINITION 1.1.2. SetH= Ha:= {H::::; G: 02(H) i-1};
and for X ~ G, set
H(X) = Ha(X) :={HEH: X ~ H}.For X ~ Y ~ G, set
H(X, Y) = Ha(X, Y) :={HE H(X): H ~ Y}.
Define He(X) (resp. He(x, Y)) as the intersection of He with H(X) (resp.
H(X, Y)).
The subgroups in H are the primary focus of our proof, so we record here the
following elementary (but important) observations: Notice that by (QT), His an
SQT-group. As G is simple and 02 (H) -=I- 1, certainly H is proper in G; hence
by (K), simple sections of subgroups of Hare in K, so that His an SQTK-group.
Then by (2) of Theorem A (A.2.1), all simple sections of Hare also SQTK-groups.
We are interested in conditions on members H of H which will ensure that
H E He. For example, in 1.1.4.6 below, we show that each member of the collection
H(T) is in He. We begin with some well known results in that spirit, which we use
frequently:
LEMMA 1.1.3. Let ME He. Then
(1) If 1 i-N :'SJ :'SJ M, then N E He.
(2) If X is a 2-subgroup of M, and XCM(X) ::::; H::::; NM(X), then HE Heand CM(X) E He.
(3) If H::::; M and B1, ... , En are 2-subgroups of H with B1 ::::; NH(Bi) for all
i::::; j and H = n~=l NM(Bi), then HE He.
PROOF. As N :'SJ :'SJ M, 02 (F(N))::::; 02 (F(M)) = 1. Thus (1) holds. If X
is a 2-subgroup of M, then NM(X) E He by 31.16 in [Asc86a], so CM(X) E He by
(1). If XCM(X)::::; H::::; NM(X), then X::::; 02 (H), so 02 (F*(H)) centralizes X,
and hence 02 (F(H)) ::::; 02 (F(CM(X))) = 1, so that H E He. Thus (2) holds,
and (3) follows from (2) by induction on n. D
For X::::; G let S2(X) be the set of nontrivial 2-subgroups of X, and let S2(G)
consist of those S E S 2 ( G) such that Na (S) E He. Here is a collection of conditions
sufficient to ensure that various overgroups and subgroups are in He:
He.
LEMMA 1.1.4. (1) If U E S2(G) and U::::; VE S2(G), then VE S2(G).
(2) If 1 -=I-U :'SJ T, then U E S2(G). In particular 2-locals containing T are in
(3) If U E S 2 (G) and 1 -=I-Z(T) n U, then U E S2(G).
(4) If 1 -=f-N::::; M::::; G with ME He and Co 2 (M)(02(N))::::; N, then NE He.
(5) If 1 -=f-N::::; ME M(T) with Co 2 (M)(02(N))::::; N, then NE He.(6) H(T) ~He.
(7) If ME He, SE Syh(M), and 1 -=I-M1 ::::; M with IS: Sn Mil ::::; 2, then