5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 645L1(G, T) using 1.2.10, contrary to 5.1.14.1. Thus 5.1.21 says [Vi, QK] = Z 1 , proving
the first assertion of (3). Hence as Vi = [V 2 , L2], L2 = [L 2 , QK]· Now K :::;! G1
by 5.1.14.2, so Ca(V2):::; G1:::; Na(QK), and hence CqAVi):::; 02(G2). ThereforeP := (CqK(Vi)a^2 ) :::; 02(G2), and [Cc(Vi), QK] ::::: CqK(Vi) ::::: P. Then L2 =
[L2,QK]::::: Ca(Cc(V2)/P), so as G2 = L2TCc(Vi), L2P :s! G 2. Then as P:::::
02(G2) :::; T ::::: Nc(L2), L2 = 02 (L2P) :s! G2. Now since L2 = D 302 (L2) with02(L2) = CL 2 (Vi), D302(Cc(Vi)) :s! G2. Therefore (3) holds, and it remains to
establish (1).
Now B acts on D3 and B :::; K::::: Ca(Z 1 ), so B centralizes (Zf^3 ) =Vi. On
the other hand as G2 is an SQTK-group, m 3 (G2) ::::: 2, so by (3), m 3 (Ca(Vi)) = 1.
Further Ca(Vi) = Ca 1 (Vi), with G1 ::::: M+. As CK is a 3'-group by 5.1.16.1, either
031 (M+) = K, or 031 (M+)/0 3 1(0^31 (M+)) ~ PGL 3 (4). In particular as Sylow
3-groups of PGL3(4) are of exponent 3 and m3(Ca(Vi)) = 1, B E Syl 3 (Cc(Vi)).
Therefore as B:::; Kand Ca(Vi) :::; G 1 ::::: Na(K), Y := 031 (Cc(Vi)) :::; K. Then
as BT is the unique maximal subgroup of KT containing BT, and [K, Vi] -/=-1,
we conclude Y = 031 (TB). Thus to complete the proof of (1) and hence of the
lemma, it remains to show X := 0{^2 ,^3 }(Cc(Vi)) = 1. As X is BT-invariant
and AutBr(K/02(K)) is maximal in AutKr(K/02(K)), X :::; CK. Therefore
(H, D3) ::::: Na(X), so if X -/=-1, then by 5.1.14.1, D3 ::::: Nc(X) :::; M+, contra-
dicting 5.1.20. This establishes (1), and completes the proof of 5.1.22. D
LEMMA 5.1.23. (v;^01 ) is abelian.
PROOF. We specialize to the case H 1 = G 1 , and recall Hypothesis G.2.1 issatisfied with L2, Vi, Z1, G1 in the roles of "L, V, Vi, H". Our proof is by
contradiction, so we assume that U is nonabelian. Then [Vi, U] = Z 1 using 5.1.21,
so L 2 = [L 2 , U], and hence the hypotheses of G.2.3 are also satisfied. So setting
I:= (Ua^2 ), G.2.3 gives us an I-series
1 =So :::; 81 ::::: 82 ::::: 83 = S := 02(I)
such that 81 =Vi, 82 =Un U^9 for g E D3 - G1, [82,I] ::::: 81 =Vi, and S/82
is the sum of natural modules for I/S ~ L2(2) with (Un S)/82 = Cs;s 2 (U). As
L 2 has at least two noncentral chief factors on V and one on (Sn L)/CsnL(V),
m := m((U n S)/82) > 1.
Let Gi := Gif Ccl CU), w := u n s, and A:= U^9 n s. Observe
B2 = AnU:::; Cu(A)
and [U, a] i 82 for each a EA - 82. Thus as Z1 ::::: 82, 82 = CA(U). Therefore as
m(U/(U n S)) = 1 since I/S ~ L 2 (2),
m(A) = m(A/8 2 ) = m((U n S)/ 82) = m = m(U /S2) - 1 2: m(U /Cu(A)) - 1,
so A* E Qr(G]', U), where r := (m+l)/m < 2 as m > 1. Let C1 := Ca 1 (K/02(K));
we apply D.2.13 to Gi in the role of "G". By 5.1.16.1, C1 is a 3^1 -group, so as
r A u ::::: r < 2, D.2.13 says that [F(Ci), A] = 1. But as G1 :::; Nc(K), F*(Gi) =
KF(Ci), so either A is faithful on K, or by 5.1.16.2, A* acts nontrivially on
a component X* ~ Sz(2k) of Ci- Let Y := K in the first case, and Y := X
in the second. By A.1.42.2 there is WE Irr+(U,Y,T); set Ur:= (WT). As
Y = [Y,A*], CA(Ur) <A. Then by D.2.7,