12.3. ELIMINATING A7 8111, and we compute that this does not hold if A= A 2. This contradiction completes
the proof of 12.3.10. DLEMMA 12.3.11. If H E 7-l(T) with n(H) = 1, then H::::; M.
PROOF. By 12.3.9.2 and 12.3.10, min{r, w} > 1, so the lemma follows from
E.3.35.1. DLet e := e(4). Since L does not appear in conclusions (2)-(4) of Theorem
12.2.13, conclusion (1) of Theorem 12.2.13 holds: Gv i. M for some v EV#. By
12.3.4, 12.3.6, and 12.3.8, we may take v = e. Thus there is HE 7-l*(T,M) with
H::::; Ge. Set K := 02 (H); as usual, Ki. M.LEMMA 12.3.12. K :::1 Ge, K/02(K) ~ A5, K = [K, J(T)], and [Z, K] is the
A5-module.PROOF. By 12.3.11 and E.1.13, H is not solvable. By 12.3.6, Ca(Z) ::::; M,
so we may apply 12.2.7.2 to conclude that K/0 2 (K) ~ A 5 • By 1.2.4, we may
embed K ::::; Ke E C(Ge) ~ C(G, T), and Ke ::; Ko E C*(G, T). As [VH, K] i:- 1
by 12.2.7.1, 1.2.9.1 says Ko E Cj(G, T). Then by 12.2.7.3, either K = Ko or
Ka/02,z(Ko) ~ A7·
Assume first that K < K 0 • Then by 12.2.7.3, Hypothesis 12.2.3 is satisfied
with Ko in the role of "L". Hence as Ko/0 2 ,z(Ko) ~ A7, the hypotheses of this
section hold with Ko in the role of L, so we may apply the results obtained so far toKa. Set Vo:= D1(Z(02(KoT))). By 12.3.7, Vo= VKEEiCz(Ko), with VK = [Z, Ko],
[Z, K] is the A 5 -module, and K = [K, J(T)]. Thus the lemma holds in this case
if K = Ke. On the other hand if K < Ke, then Ke = Ko by A.3.14. Further by12.2.8, Ko contains all elements of order 3 in Ge, so in particular Le ::::; Ko. But
K is the unique member of C(KoT, T) with K/02(K) ~ A 5 , so K = L~ ::; M,
contrary to K i. M.
Thus we may assume that K = Ko = Ke E C(G, T). Therefore Ge ::;
Na(K) = !M(KT) by 1.2.7.3. Then there is H1 E 7-l(T,Na(K)), and in par-
ticular H1 i. Ge. Thus [Z, H1] ¥:-1, so K = [K, J(T)] and [Z, K] is an FF-module
by Theorem 3.1.8.3. By 12.2.7.3, [Z, K] the sum of A 5 -modules, and then by The-
orem B.5.1.1, [Z, K] is an A 5 -module, completing the proof of the lemma. D
Next by 12.3.4 and 12.3.7.1, C.Mv(e) = M1 x Nh where M 1 ~ 84 is the
pointwise stabilizer in Mv of {1, 2, 7}, and M2 ~ S 3 is the pointwise stabilizer
of {3, 4, 5, 6}. Let Li := 031 (Mi), so that Le = L 1 L 2 , and Ll = 031 (CL(Z n
V)) = 0
31
(CL(Z)) using 12.3.7.1. Let P E Syl3(Le)· By 12.3.12, K :::1 Ge, so
P = (PnK) x Cp(K/02(K)), and hence P '¥5- 31 +2. Therefore 02,z(L) = 02(L),
and appealing to 12.2.8:
LEMMA 12.3.13. L/02(L) ~ A7 and L = 031 (M).
We are now in a position to complete the proof of Theorem 12.3.1. As L =
031 (M) by 12.3.13 and Ca(Z) ::; M by 12.3.6, L 1 = 031 (CL(Z)) = 031 (Ca(Z)).
By 12.3.12, [Z, K] is the A 5 -module, so 02 (KnM) centralizes Zn [Z, K], and hence
02 (K n M) centralizes Z by B.2.14. Thus Li = 0
31
(Kn M). Then as Li and L 2are the T-invariant subgroups X = 02 (X) of Le with IX : 02 (X)I = 3, it follows
that L2 = 02 (CLe(K/02(K)).