704 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW7.7.2. More detailed properties of V 0 and its centralizer. Observe
C.M(Vo) is the subgroup of L fixing 1 and acting on the subspace (2, 3), so C.M(Vo) ~
Lz(2) ~ 83.
Set L^0 := 02 (CL(V 0 )), so that L^0 /0 2 (L^0 ) is of order 3. Let() E L^0 be of order- Observe
andV =Vo E9 V_.
Set T 0 := CT(Vo) and Mo:= CM(Vo). Then CLT(Vo) = L^0 To, To E Syl2(CM(Vo)),
and To of order 2 is generated by the involution I defined in the previous subsection.
Let Z1 := (1), Gi := Ca(Z1), and Li := 02 (CL(Z1)). Thus Z1 ::; Vo, so
Go ::; G 1 and L^0 ::; L 1. Again Li/0 2 (L 1 ) is of order 3, but Li/Q ~ A4 whileL^0 /Q ~ Z3.
Let V+ denote either V 0 or Z1, and define G+ := Ca(V+), L+ := 02 (CL(V+)),
M+ := CM(Z+), and T+ := CT(V+)· Then
M+ = CM(V)L+T+,
and by 3.2.10.4, T+ is Sylow in G+.We emphasize that
and that this property is crucial to our proof that Go ::; M.
LEMMA 7.7.4. IfY is an abelian subgroup of CM(V+) of odd order, then
(1) Yo := Cy(V) is of index at most 3 in Y, and
(2) if Yo=!= 1, then Na(Yo)::; M.
PROOF. As Y is of odd order in 02 (CM(V+)) = 02 (CM(V))L+ and IL+02(L+)I = 3, IY: Yol::; 3. By Theorem 4.4.3 and Remark 4.4.2, Na(Yo)::; M. D
LEMMA 7.7.5. If w EV# is 2-central in G, and L+T+::; H::; G+, then
PROOF. We show that the hypotheses of 1.1.4.4 are satisfied with Gw := Ca(w)
in the role of "M", and H n Gw in the role of "N". First Gw E He by 1.1.4.3 and
our hypothesis that w is 2-central. Set G+,w := Ca(V+(w)), and embed Q ::;
Tw E Sylz(G+,w)· Then J(T) ::; Q ::; Tw so Tw ::; Na(Tw) ::; M by 3.2.10.8.
Consequently Tw ::; M+, which we saw above is CM(V)L+T+. 'Then by Sylow's
Theorem, T;}, ::; L+T+ for some c E CM(V) ::S: G+,w, so without loss Tw ::S: L+T+ ::;
H. Hence ll.+::; H n 02 (G+) n Gw::::; 02 (H n Gw)· SoCo 2 (aw)(02(H n Gw))::; Co 2 (Gw)(V+)::; 02(Gw) n G+,w::; 02(G+,w)
::; Tw ::; H n Gw.
Thus we finally have the hypothesis for 1.1.4.4, and we conclude from 1.1.4.4 that
HnGw E He. D