12.3. ELIMINATING A 7 809{3) Mv ~ 81 or A1, and form= 2, 4, 6, CMv (e(m)) is isomorphic to Z 2 x (^85)
or 85; 84 x 83 or a subgroup of index 2 in 84 x 83; 85 or A5; respectively.
LEMMA 12.3.5. L controls fusion of involutions in V.
PROOF. Recall Na(T) = NM(T) by Theorem 3.3.1, and this subgroup controls
fusion of involutions in Z by Burnside's Fusion Lemma A.1.35. We saw in 12.3.4.3
that the three involutions in Zn V are not Mv-conjugate; hence they are not M-
conjugate since V is a TI-set in M by 12.2.6. Further by 12.3.4, each member of V
is fused into Z n V under L, so the lemma holds. DLEMMA 12.3.6. Ge(6):::; M.
PROOF. Let e := e(6). By 12.3.4, 02(Le'f') = 1 and Le ~ A 6 , so applying
12.2.12, Hypothesis C.2.3 is satisfied by Gv, Mv, Q. Also Le/02(Le) ~ A 6 or
A 6 for L/02(L) ~ A 7 or A1, respectively, so Le E C(G, T) and hence Le :::; K E
C(Ge) ~ C(G, T).^4 As Le involves A5, K/02(K) is quasisimple by 1.2.1.4; further
Ge E 1ie by 1.1.4.2, so that KE 1ie by 1.1.3.l. Then if Le< K, Kand KnM aredescribed in the list of conclusion (3) of Theorem C.2.7; but we find no case where
Kn M contains a T-invariant subgroup Le with Le/02,z(Le) ~ A5.
Thus Le = K. Now B(Ge) = Le by A.3.18, so B(Ge) :::; M. Set Y :=
Ca.(K/02(K)); then Q E 8yb(Y) by 12.2.12.1. Thus Ge= YNa.(Q) = YMe
by a Frattini Argument. Further Hypothesis C.2.3 is satisfied with Y, Y n M, Q in
the roles of "H, MH, R"' so by C.2.5, y is the product of y n M with xo-blocks.
Hence as each xo-block is generated by elements of order 3, Ge = B(Y)Me :::; M,
completing the proof. D
LEMMA 12.3.7. (1) L = [L, J(T)], so Mv ~ 81 and f"h(Z(02(LT))) = V EB
Cz(L).
(2) Let Ke:= L'J_ 2 ). Then Ke= [Ke,J(T)] and Sl1(Z(02(KeT))) = [V,Ke] EB
Cz(Ke)·
PROOF. By 12.3.6, Ca(Z) :::; M, so L = [L, J(T)] by 12.2.9.2. Hence by
B.3.2.4, Mv ~ 87 , and if A E A(T) with A-/= 1, then A is generated by transvec-
tions and m(A) = m(V/Cv(A)). In particular Ke= [Ke, A] for some such A. Let
Zx := S11(Z(02(X)) for X := LT or KeT. As m(A) = m(V/Cv(A)), ZLr =
VCzLT(A), so V = [ZLr,L]. Then as the 1-cohomology of V under L/02(L) ~ A1
is trivial by I.1.6.1, ZLr = VEBCzLT(LA). Hence as LT= LA02(LT), CzLT(L):::;
CzLT(T) :::; Z, and (1) follows. Similarly Z:K.r = [V,Ke] EB Cz(Ke), so that (2)
holds. D
. LEMMA 12.3.8. Ge(2) :::; M.
PROOF. Let e := e(2) and Ke:= L';'. Then Ke:::; KE C(Ge) ~ C(G,T), and
K:::; Ko E C*(G,T). As Ke E C1(G,T), Kand Ko are also in Ct(G,T) by 1.2.9.1,
and Ko E Cj(G, T) by 1.2.9.2. Let Vo := S11(Z(02(KoT)). Then e, e(6) E Z:::; Vo
since F*(K 0 T) = 02 (K 0 T) by 1.1.4.6, so [V, Ke] = [e(6), Ke] :::; Vo, and hence
[Vo; Ko]-/= 1. By 12.3.7.2, Ke= [Ke, J(T)], so Ko= [Ko, J(T)].
Suppose K 0 /0 2 (K 0 ) is not quasisimple. Then Ke < Ko, so as Ke/02(Ke) ~
A 5 , the embedding of Ke in Ko is described in cases (13) or (14) of A.3.14. As
(^4) Just as for McL in 12.3.3, the shadow of n1(3) is now ell~inated by the application of
1.2.9.1, as in that group K would be n6(3).