1.1. THE COLLECTION 'He 501PROOF. Assume the hypotheses of (1) and set N := Na(U). Then by hypoth-
esis NE He. Now if U :::'.) V then VS N, so NN(V) E He by 1.1.3.2. But
02 (F*(Na(V))) s Ca(V) S Ca(U) s N,
so 02 (F(Na(V))) S 02 (F(NN(V))) = 1 as NN(V) E He. Therefore Na(V) E
He as desired. This shows that (1) holds when U :::'.) V. Then as U :::'.) :::'.) V, (1)
holds by induction on IV: UJ.
Under the hypotheses of (2), Na(U) is contained in some X E M(T), and,
as we remarked earlier, X E He by Hypothesis (E). Then as Na(U) = Nx(U),
Na(U) E He by 1.1.3.2, proving (2).For (3), observe Z(T) n U E Si(G) by (2), and then U E Si(G) by (1).
Now assume the hypotheses of (4) and set R := Co 2 (M)(02(N)). As RS NS
M by hypothesis, we conclude RS 02(N); and then 02 (N) and Rare centralized
by 02 (F*(N)) =: L. Then as L = 02 (L), the Thompson Ax B-lemma A.1.18 says
L centralizes 02(M). But 02(M) = F*(M) as ME He, so that LS Z(0 2 (M)),
and then L = 02 (L) forces L = 1. Thus (4) is established.As G is of even characteristic, M(T) ~He, so (4) implies (5).
If NE H(T), then 02 (N) i= 1, so there is M such that
TS NS Na(02(N)) SM E M(Na(02(N))).
Then as TE Syh(M), ME He by (E), and also 02 (M) SN by A.1.6. Therefore
NE He by (5), proving (6).Finally assume the hypotheses of (7) and set M2 := Mi02(M). By (4), M2 E
He. But as JS: Sn Mil S 2, JM2: Mil S 2, and so Mi :::'.) M2. Then Mi E He by
1.1.3.1 establishing (7).
This completes the proof of 1.1.4. DWe also need to control members of H which are not in He. The following resultgives some control in an important special case. For example, the subsequent result
1.1.6 shows that the hypotheses are achieved in any sufficiently large subgroup of
. a 2-local subgroup.
Recall our convention in Notation A.3.5 that A5, A7, and M22 denote the
nonsplit 3-fold covers of A5, A1, and M22·LEMMA 1.1.5. Let HEH, 8 E Syl2(H), and ME He(S). Assume that
Co2(M)(02(H n M)) s H,
and ME H(Ca(z)) for some 1 i= z E [h(Z(S)). Then:(1) F*(H n M) = 02(H n M).
(2) z inverts O(H).
(3) If L is a component of H, then L = [L, z] i M and one of the following
holds:(a) L is simple of Lie type and characteristic 2, described in conclusion
(3) or (4) of Theorem C (A.2.3), and z induces an inner automorphism on L.
(b) 1 i= Z(L) = 02 (L) and L/02(L) is L3(4) or G2(4), with z inducing
an inner automorphism on L.