16.4. INTERSECTIONS OF Na(L) WITH CONJUGATES OF Ca(L) 1189We will first show in 16.4.9.2 that Ao is nonempty. The shadow of 810 is eliminated
toward the end of the proof: that is, a transposition in 810 is a 2-central involution
· with centralizer Z2 x 8s, such that Ao = 0; of course 810 is neither simple nor
quasi thin.LEMMA 16.4.9. (1) If K' tJ. Ao, then IRI = 2.
(2) Ao -/:-0.
(3) Assume J E K^0 , and some involution i in J induces a nontrivial inner
automorphism on L. Then J E Ao.PROOF. First assume (1) and the hypothesis of (3). Then i E LK - K, so
J-/:-Kand INJ(K)l2 > 1, and hence J EA. Now if INJ(K)b > 2 then J E Ao
since we are assuming (1), while if INJ(K)b = 2, then (i) E 8yh(NJ(K)) withi E LK, so that J E Ao by definition. Thus (1) implies (3), so it remains to
establish (1) and (2).
If A= Ao then (1) is vacuous and (2) holds as A is nonempty, so we may assume
K' E A - Ao and pick some involution r E R inducing an outer automorphism onL. Then by 16.4.8, Lr :::; L' :::; Ca(R).
We now prove (1). By inspection of the centralizers of involutory outer auto-
morphisms of L* listed in 16.1.4 and 16.1.5, one of the following holds:(I) CH (L;) = (r).
(II) LT ~ 88 and r* is of type 23, 12.
(III) L* ~ M12 and L; ~As.
In case (I), as R* centralizes L; and R ~ R*, R = (r) is of order 2, and hence (1)holds in this case. Thus we may assume that (II) or (III) holds. In either case,
CH (L;) ~ E 4 , so either R is of order 2 and (1) holds, cir R =CH* (L;) ~ E 4 , and
we may assume the latter.. Suppose case (II) holds. Then there is s E R# with s* of type 2, 16. But then
[R*, L~] -/:-1, a contradiction since Ls :::; L' :::; Ca(R) by 16.4.8.
Therefore case (III) holds. Then there is s E R# with s E L but s* not
2-central in L*. Let SL denote the projection of son L. If 02(L)-/:-1, then from
I.2.2.5b, BL is of order 4, so s = SLSC with sc E NT 0 (K') of order 4, impossible as
NT 0 (K') ~ R ~ R ~ E4 by 16.4.2.1. Therefore 02(L) = 1, and hence CL(s*) =
CL(s)*, with CL(s) :::; Na(K') by 16.4.2.5. Thus (sL) = [CTL (s), r] :::; T n K' = R
by 16.4.2.3, and hences= BL EL. ThenTc= CT 0 (s) :S: NT 0 (K^1 ) = CT 0 (R) ~ R ~ E4,
so Tc~ E 4 centralizes R. Then as L*(r*) = Aut(L*), T = TcTLR:::; Ca(Tc) soTc :::; Z(T). Hence R is in the center of each Sylow 2-subgroup of H' containing R.
As CT(s):::; H' and [R, CTL (s)] -/:-1, this is a contradiction. Thus (1) is established.
We may assume that (2) fails, and it remains to derive a contradiction. Now
R = (r) is of order 2 by (1). By 16.4.4.1, CTa (r) = (z) is of order 2, and TcR isdihedral or semidihedral. Set E := (r, z). By (1) and (3):
If J EA, then INJ(K)l2 = 2 and Jn KL is of odd order. (*)
By 16.4.3.1, KE A(K'), so we have symmetry between Kand K'. Thus applying
(*) with the roles of K and K' reversed, we conclude that z induces an outerautomorphism on L'.
Next we show: