1190 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC
(+) If r^1 = rv for some l EL and 1 i= v E .CL(r), then r^1 acts on K' with
(r) E Syl2(CK1(r^1 )) and v f: L'.
For assume the hypotheses of ( +), and let J := ( K')^1 • Then r^1 E CJ ( r) :::; NJ ( K'),
and as R has order 2, r^1 ¢: K', so that J "I-K'. Thus J E f:..(K'), and then
by (*), R E Sylz(NK1(J)), and also Jn K'L' is of odd order so that v ri L'.
As (r) = R E Syl 2 (NK'(J)) and Cc(r^1 ) :::; Nc(J), also (r) E Syl2(CK1(r^1 )),
completing th~ proof of ( +).
As r induces an outer automorphism on L, by 16.1.6, r does not centralize TL
unless Lis A5. In the first case, there is l E TL with r-/= r^1 E Cc(r), and in the
second by inspection there is l E L with this property; thus in any case there is
l EL with r "I-r^1 E Cc(r). Let J := (K')Z.
Suppose that Tc > (z). Then r^1 normalizes some subgroup ·X == (x,r) of
order 4 in K'; and (r) E Syl 2 (CK'(r^1 )) by ( + ), so that (r^1 , X) ~ Ds, and hence
v := rr^1 = rlx E JX n L. Thus JX ¢: t:.. by (), so JX = K and hence v E
Kn L = Z(L), so z = v EL as (z) = CT 0 (r). Further rL n Ca(r) = {r,rz},
so ICL·(r) : CL(r)I = 2 and rCL(r) n rL = {r}, so there· are involutions
in CL•(r) - CL(r). Suppose L* ~ L3(4) or Mzz. Examining 16.1.4 and 16.1.5
for outer automorphisms r with CL (r) not perfect, we conclude that either L ~
L 3 (4), r is a graph-field automorphism, and CL (r) ~ Qs/Eg; ·or L* ~ M22 and
CL (r) ~ Z 4 /Z 5 / E 2 4. However in both cases each subgroup of CL (r) of index
2 contains all involutions in CL (r), contrary to an earlier remark.
We have shown that either (z) =Tc E Syl 2 (K), or z EL and L* is not L3(4)
or Mzz...
As r induces a~ outer automorphism on L, Lr :::; L' by 16.4.8. So if rv = r^1 for
some l EL and 1 "I-v E Lr, then v EL' which is contrary to ( + ). Thus:
(!)It is now fairly easy to eliminate most possibilities for the involutory outer
automorphism r on L in 16.1.4 and 16.1.5; indeed the next few paragraphs will be
devoted to the reduction to the following cases:
(i) L ~ A5 or As..(ii) L ~ L3(4), and r induces a graph-field automorphism on L*.
We may assume that neither (i) nor (ii) holds, and will derive a contradiction. Since
r induces an outer automorphism on L, Lis not L 3 (2) by 16.4.6. Then as (ii) does
not hold, by inspection of the outer automorphisms in the remaining cases in 16.1.4
and 16.1.5, Lr is of even order. Choose notation so that CT(r) E Sylz(CH(r)).
Suppose first that L ~ Mz2 or HS, and let Tr := T n RCL(r)); then Z(Tr) =
RX Zr, where Z2 ~Zr= (v) :::; Lr. So as Tr <RTL E Sylz(RL), rv E rNTL(Tr),
contrary to (!). Thus if L* ~ Mz 2 or HS, we may assume that 02 (L) "I-1.
Now assume that 02(L) = 1, and recall L is not A 6 , As, M 22 , or HS by
assumption. Therefore by inspection of the outer automorphisms in the possibilities
remaining in 16.1.4 and 16.1.5, Lis transitive on the involutions in r L. Then since
Lr is of even order (recalling L * ~ L 3 ( 4) as we are assuming that (ii) fails), (!)
supplies a contradiction.
Thus to establish our reduction, it remains to treat the case 1 "I- 02 (L) = Z(L).
Since L admits the outer automorphism r of order 2, we conclude from 16.1.2.l
that L* is L3(4), G2(4), M12, Mz2, Jz, or HS. If ITcl = 2, then (z) .= Z(L). If
;~