502 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS(e) L ~ £ 3 (3) or L 2 (p), p a Fermat or Mersenne prime, and z induces an
inner automorphism on L.
(f) L/0 2 (L) is a Mathieu group, J 2 , J4, HS, He, or Ru; and z induces a
2-central inner automorphism on L.
PROOF. Part (1) follows from 1.1.4.4 applied with H n Min the role of "N",
in view of our hypothesis.
Next C 0 (z) ~ M by hypothesis, soCo(H)(z) ~ O(H) n M ~ O(H n M) = 1
by (1), giving (2).
Now assume Lis a component of H. If L ~ M then L ~ E(H n M), contrary
to (1). Thus L 1:. M so in particular L 1:. Co(z).
As z E Z(S) and S E Syb(H), z normalizes each component of H; so as
L f:. Co(z), L = [L,z].
Set R := Ns(L) and (RL)* := RL/0 2 (RL). Then R E Syb(RL) so R* ESyl 2 (R L) and z is an involution in the center of R. By hypothesis, Co(z) ~ M,
so CH(z) = CHnM(z). Now H n M E 1-le by (1), so by 1.1.3.2, CHnM(z) E 'He.
Since L :::1 :::J H we haveCL(z) :::1 :::1 CH(z) = CHnM(z),
so CL(z) E 1-le by 1.1.3.1. Also 02 (CL(z)) = 02 (CL(z)) by Coprime Action,
while 02(RL) n L ~ 02(L) ~ Z(L), so we conclude F(CL· (z)) = 02(CL (z*))
from A.1.8.
If z induces an inner automorphism on L then z centralizes Z(L), so O(L) = 1
by (2), and hence Z(L) = 02 (£). Put another way (recalling Lis quasisimple), if
O(L) =f=. 1 then z induces an outer automorphism on L.
As His an SQTK-group, we may examine the possibilities for L/Z(L) appear-
ing on the list of Theorem C.Suppose first that L/Z(L) is of Lie type and characteristic 2; then L* appears
in conclusion (3) or (4) of Theorem C. Now z E Z(R), so from the structure of
Aut(L), either z EL, or L is A 6 or A 6 with z inducing a transposition on L.
However in the latter case as 02 (£) = 1, or else L/O(L) ~ SL 2 (9) by I.2.2.1, so
that the transposition z does not centralize a Sylow 2-subgroup of L, contrary to
z E Z(R); hence (c) holds. Thus we may assume z EL, so by an earlier remark,
O(L) = 1. Thus either Z(L) = 1, so Lis simple and (a) holds; or from the list of
Schur multipliers in I.1.3, L* is £ 2 (4), A 6 , Sz(8), L 3 (4), G 2 (4), or £ 4 (2). Then as
z centralizes a·Sylow 2-group of L, when L* ~ L 2 (4) ~ A 5 , A 6 , or Sz(8), we obtain
a contradiction from the structure of the covering group Lin (1) or (4) of I.2.2, or
in 33.15 of [Asc86a]. This leaves covers of £ 3 (4) and G 2 (4), which are allowed in
(b).
We have shown that the lemma holds if L/Z(L) is of Lie type and characteristic
2. But A5 ~ 0,4(2), A5 ~ Sp4(2)', and As ~ L4(2), so if L* is an alternating
group, then from conclusion (1) of Theorem C and I.1.3, L ~ A 7 or A7. As
F(CL(z)) = 02(CL·(z)), z tj:. L and z is not a transvection, so we conclude
z* has cycle structure 23. As z centralizes a Sylow 2-group of L, we conclude that
02(L) = 1, from the structure of the double cover of A 7 in 33.15 of [Asc86a]. So
(d) holds.