546 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1
of order 27 and rank 4, and Hu =Na,,. (U) ~ Ds x S4, to conclude that K ~ A5,
L 2 (7), or L 2 (17). Next 02 (Na.,,.(U)) = 02 (Hu) ~ A4 and 02 (NK(U)) ~ A4 in
each of the possibilities for K, so 02 (Hu) ~ K. Now z E (u)0^2 (Hu) ~ (u)K, but
z /= u, so Tu normalizes the component K, ·and hence K :SI Gu by 1.2.1.3. As
J(Tu) = R = uux, K is not L 2 (17), and in the remaining two cases, x induces an
outer automorphism on K interchanging the two 4-groups in Rn KE Syl2(K), so
that K = (0^2 (Hu), 02 (Hur). Also z = uu^8 for s ES -R, and U^8 E 02 (Hu) ~ K;
so as K has one class of involutions, by a Frattini Argument,
Gu= KCau(u^8 ) = KCa.,,.(z) = KMu = KTu-
Let D := CR(0^2 (Hu)) and UD := D n U. Then D ~ Ds, UD is a 4-group, and
UD - (u) <;;;; zL from an earlier remark. Hence as Ca(z) is solvable, Cuv(K) = (u).
But if K is L 2 (7), then CAut(K)(0^2 (Hu)) = 1, so we conclude that K ~ A5 and
v E UD - (u) induces a transposition on K. As Gu = KTu and K is simple,
B := Cau(K) = Cr.,,.(K) ~ Cr.,,.(0^2 (Hu)) = D, so B = CD(K) is of order 4, with
Gu/B ~ Aut(A 6 ), since x interchanges the two 4-groups in RnK. As R = uux,
x also interchanges the two 4-groups in R/(RnK) = D(RnK)/(RnK) ~ D, and
hence B ~ Z4, since UD 1:. B.
Let I:= 02 (M). Then I= IiI2 with Ii~ SL2(3) and [Ii,h] = 1. Further
there exists y E T - S centralizing Ii with y^2 E Z: namely any y inducing an
orthogonal transvection on P centralizing Ii. Moreover each t E T - S with t^2 E Z
is conjugate under M to y or ya, where a E Ii n P is of order 4, and exactly one
of y and ya is an involution. Thus M is transitive on the set I of involutions in
M - IS, and either y or ya is a representative i for I. Let j := ia; then j2 = z.
Observe jG n S = 0: For if j9 E S then z9 = (j9)^2 E <l?(S) ~ P. But as u E P andu ¢. z^0 while M is transitive on the involutions in P - Z, Z is weakly closed in P
with respect to G; so z = z9 and hence g E Gz = M, contradicting IS :SI M.
As jG n S is empty but G = 02 ( G) with [T : S[ = 2, we can apply Generalized
Thompson Transfer A.1.37 to j in the role of "g", to see that j^2 = z must have
a G-conjugate in T - S; so i = z^9 for some g E G. Now if y = i then SL 2 (3) ~
Ii~ Ca(i) = M9, so z E 02 (Ii) ~ 02(M9) = P9. However we saw in the previous
paragraph that z^0 n P = {z }, so z = z9 = i, contradicting i tf_ S. Therefore y is of
order 4 and i =ya centralizes be, where b E I 2 is of order 4 and inverted by y, and
02(I1) =(a, e). As be E uM, we may assume be= u, so that u centralizes i. Theni E Tu - S acts on K ~ A 6 • As S 2 UD and v E UD - (u) induces a transposition
on K, KS induces the 85-subgro{ip of Aut(K) on K, so as i ¢. S, i does not induce
an automorphism in S5. Then as i is an involution, i induces an automorphism in·
PGL2(9) rather than M 10 , and hence CK(i) ~ D 10. This is impossible as i E zG
and M = Ca(z) is a {2, 3}-group. The proof of 2.4.29 is complete. D
By 2.4.29, we have reduced to the case where L 0 is the product of two A 3 -
blocks. Henceforth we let s denote an element of S ~ N 8 (L). Thus H = L 0 S and
Lo = L x L^8 • Let U1 := U and U2 := us.
LEMMA 2.4.30. (1) QQx = R = Baum(S) = J(S) for x ET - S.
(2) HE r*.(3) {Q, Qx} are the S-invariant members of A.(R).
(4) RLo = Cs(Lo) x Lout) with <l?(Cs(Lo)) = 1 and Lout)~ S4 x S4.