686 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONALK* is the direct product of copies of GLm(2) for some m 2: 3. Next as V <I T,
1 =f-V n Z(T), so by G.6.4.4, K* = GL(U). By 6.2.9,
CH·(Zs) = M; = NM(V)*.
_Thus as V is a 4-group we conclude m(U) = 3 and H f::! L3(2).
As ~(Zs)= 1 and H is transitive on tJ#, ~(U) = 1, so U f::! Em. As V is
the group oftransvections with center Zs, Zs= Co(V*), so Zs= Cu(V). Further
u:::; Cr( Zs)= TLCr(V), where TL:= T n L; thus 101 = IU/Cu(V)I = IU: Zsl =
4 = lf'LI, sou= TL E Syl2(L).Now [CH(U), VJ:::; Cv(U):::; VnQH by 6.2.10.2, and we saw that VnQH:::; U.
Hence K = (VH) centralizes CH(U)/CH(U). Next CH(U)/CH(U) is a subgroup
of the group X of all transvections on U with center Z, and U is the dual of X as amodule for CcL(U) (Z). Thus as U is the natural module for K* and K centralizes
CH(U)/CH(U), we conclude CH(U) = CH(U).
Next L = [L, U] with [U, 02(LT)] :::; Cu(V) = Zs :::; V, so L is an L2(4)-block. Also Cr·(V) = V as V is a 4-subgroup of H f::! L3(2); thus Cr(V) ::;
VCr(U). Therefore as Cr(U) = Cr(U) by the previous paragraph, we conclude
Cr(V) = VCr(UV). Then as U E Syh(L), it follows from Gaschiitz's theorem
A.1.39 and C.1.13.a that L02(LT) = LCr(L). On the other hand, Cr(L) = 1
by 6.1.6.1. Therefore V = 02(LT) = 02(M) using A.1.6. Then TL = J(T) with
A(T) = {A1, A2} and Ai = V, so as m(U) = 4, U = A2· Thus as NL(TL)
acts on V, it also acts on U, so that L 0 := (NL(TL), H) acts on U, and hence
Lo := Lo/CLo(U) ::; GL(U) f::! As. As NL(TL) is transitive on zt and H is
transitive on U - Z, L 0 is transitive on U#. Further Ct 0 (z) = fI f::! L 3 (2), so we
conclude Lo f::! A7. Moreover setting Mo :=Mn Lo, NLo (Zs) ::; Mo <Lo by 6.2.1.The stabilizer of any 4-subgroup of U in L 0 is the global stabilizer in Lo of 3 of the
7 points permuted by Lo in its natural representation, which is a maximal subgroup
of Lo. Thus Mo = NL 0 (Zs). Now we can also embed T::; Y::; Lo with Y f::! Ss
and IY: Y n Mol = 5. Thus YE 1-i*(T, M) with n(Y) = 2 by E.2.2, contradicting
Hypothesis 6.1.1.2. DLEMMA 6.2.12. (1) 02 (H n M):::; CM(V):::; CM(L/02(L)).
(2) 02 (CH(U)) = 1, so CH(U) = QH.PROOF. As V* has order 2 by 6.2.11, we conclude from 6.2.9 and 6.2.2 that
H n M acts on the series V > Cv (U) > Zs > Z, and all factors in the series are of
rank l. Therefore 02(H n M) centralizes V by Coprime Action. Then 02 (H n M)
centralizes L/0 2 (L), proving (1).
Next using 6.2.10.2 and (1), X := 02 (CH(U)) :=:;: 02 (H n M). Thus X cen-
tralizes L/02(L), so that L normalizes 02 (X0 2 (L)) = X. Now if X =f. l, then
02(X) =f-1by1.1.3.1, since HE 7-ie by 1.1.4.6. But then H:::; Nc(0 2 (X)) :=; M =
!M(LT), contradicting H i M. This shows that CH(U) is a 2-group, and then
6.2.10.1 completes the proof of (2). D
We can now isolate the case leading to M 22 , which we identify via a recent
characterization of Chao Ku. Recall that U = (Zf), so that Zs:::; V n U.
PROPOSITION 6.2.13. If Zs= V n U, then G f::! M22·