2.4. THE CASE WHERE r 0 IS NONEMPTY 543
S1 E Syb(X) with T1 ::; S1, we have L E .C(X, S 1 ). This is a contradiction,
since from the 2-local structure of the groups K on our list, none contains an A 5 -
block L, such that for each overgroup X of LS+ in K with F*(X) = 02 (X) and
8+ E 8yl2(NK(L)), L E .C(X, 81) for 8+ ::; 81 E 8yl2(X). This completes the
proof of 2.4.25. · D
By 2.4. 7 and 2.4.25, either L is an A5-block with s = 1, or L is an A 3 -block
withs= 1 or 2. So by 2.4.5.3, the hypotheses of Theorem C.6.1 are satisfied with
T, Sin the roles of "A, TH". Similarly by 2.4.5.1, we can appeal to results from
section C.5, with S, S, Lo, Uo, Baum(S) in the roles of "TH, R, K, U, 8".
We will first show that whens= 1 and Lis an A 3 -block, then G is a group in
the conclusion of Theorem 2.4.1. Since G is a counterexample to Theorem 2.4.1,
this will establish the following reduction:
LEMMA 2.4.26. If L is an A 3 -block, thens= 2.
PROOF. Assume L is an A3-block with s = 1. By Theorem C.6.1, H ~ 84 or
Z2 x S4.
Suppose first that H ~ S4, so that case (b) of Theorem C.6.1.6 holds, and in
particular Tis dihedral or semidihedral. Then by I.4.3, G is L 2 (p), pa Fermat or
Mersenne prime, A5, L3(3), or Mu. As M = !M(T), G is not L 2 (7) or A 6 • As
o-=/:-0, G is not L2(5). This leaves the groups in Theorem 2.4.1, contradicting the
choice of H, Gas a counterexample.
Therefore H ~ Z2 x S4, so case (a) of Theorem C.6.1.6 holds. Then IT: SI= 2
and J(S) = S = J(T). By C.6.1.1, S = QQx for x E T - 8. Define y and z
by (y) = Z(H) and (z) = <P(S); by 2.4.4, 8 = CT(y). Since 8 = J(T) is weakly
closed in T, by Burnside's Fusion Lemma A.1.35, Na(S) controls fusion in Z(8),
so y tJ. z^0. Thus yx = yz, and H is transitive on yU - {y}, so all involutions in
yUUx are in y^0 , and all involutions in uux are in z^0.
Suppose first that y^0 nT ~ S. Then y^0 nT ~ yUUx. Now T/UUx is of order
4 and hence abelian, so by Generalized Thompson Tranfer A.1.37.2, y tJ. 02 (G),
contradicting the simplicity of G.
Thus we may take x E · y^0 ; in particular, x is now an involution. Let u E
U - (z). Then (u, x) ~ D15, and we saw [x, y] = z, so 81 := (xy, u) ~ SD 16 ,
with xy of order 4. Hence all involutions in S 1 are in uux and therefore lie in z^0.
Therefore y^0 n 8 1 = 0, so Thompson Transfer produces our usual contradiction to
the simplicity of G, completing the proof. D
By 2.4.25 and 2.4.26, the structure of S is similar in the two remaining cases
where Lo is either an As-block or the product of two A3-blocks; we summarize some
of these commo.n features in the next lemma:
LEMMA 2.4.27. (1) IT: SI= 2 and R = Baum(8) = J(S) = J(T).
(2) A(T) = {Q,Qx,A1,A1} for x ET-8, r E S-R, and IA1: Ai n QI= 2.
(3) Let Tc := CT(Lo). Then <P(Tc) = 1, Q =Tc x Uo, and Tc n T 0 = 1 for
each x ET-8.
(4) R =Tc x UoU 0 , with Lo= [Lo, U 0 ].
PROOF. Let M 0 := NT(S); by Theorem C.6.1, IMo: SI= 2. Thus by 2.4.5.2,
the hypotheses of C.5.6.7 are satisfied. Further by C.6.1.1, QQx = R = Baum(8) =
J(S). By 2.4.25 and 2.4.26, Lo is an A5-block or the product of two A3-blocks, so
by C.6.1.4, A(R) = A(S) is described in (2). Thus to complete the proof of (2),
