12.6. ELIMINATING As ON THE PERMUTATION MODULE S27remains to establish (2), we may assume that L is a block. Let a := g-^1 Notice
that V :::! R, so that also va :::! R.
Suppose first that [V, vaJ = 1. Then va :::; CR(V) = Q, so as L is a block,
[VVa,LJ:::; [Q,LJ:::; V:::; vva. Similarly [VVa,Kv]:::; vva. Therefore as LR=LT, Kv:::; M = !M(LT), contrary to 12.6.11.
Thus [Va, VJ# 1, so as va:::; R = Rv, va = ((1, 2)), and hence [V, vaJ = (e 1 , 2 ).
Since Zv # 1 by hypothesis, vis chosen to have weight 2, so v = e 1 , 2. By symmetry,
[Va, VJ = (va), so v = va = vg-
1
, and hence g E Gv; impossible as vg centralizesL. 0
LEMMA 12.6.17. Na(Rv) f:. M.
PROOF. Assume that Na(Rv) :::; M. Then as Rv E Syl2(CM(Lv/02(Lv))),
Rv = R. Hence NK;;(R) = NKJRv):::; M;, so R* # 1 as Kv f:. M by 12.6.11.
In view of 12.6.15, K; appears in the list of Theorem B.4.2, so since KZS* has a
nontrivial 2-subgroup R such that L~ :::] N K;; (R) ~ T; with L~/0 2 (L~) ~ A 6 and
IS* : T; I :::; 4, we conclude that KZ S* ~ Ss and R* induces a transposition on KZ.
Then IS*: T;I = 4 =IT: Tvl, SOS E Sylz(G). By 12.6.13.3, Vv:::; [U,Kv] =: Uv.
Then from Theorem B.5.1.1, Uv/Cuv (Kv) is the 6-dimensional quotient of the core
of the permutation module for KZ. Further [ Cvv ( Lv), Kv J # 1, as v 0 is of weight
6 in both Uv and V. But vis central in Gv, so that v tj. Cvv(Lv) = (vo), and
hence Zv # 1. Now 12.6.16.1 supplies a contradiction, completing the proof of the
lemma. 0LEMMA 12.6.18. (1) L is an As-block.
(2) Kv is an A1-block.
(3) Zv # 1.
(4) LT= LRv.PROOF. By 12.6.17, Na(Rv) 1:. M. So as Na(Q) :::; M, Q < Rv, and hence
Rv = ((1,2)), so (4) holds. Then Rv:::; X:::; M for some X ~ S 3 -so either thereis 1 # C char Rv with C :::] X, or we may apply C.1.29 to Rv E Sylz(X), to
conclude that 02 (X) is an A 3 -block. In the former case, C :::] (X,CLr(v)) =LT,
so Na(Rv) :::; Na(C) :::; M = !M(LT), contrary to 12.6.17. Thus X an A 3 -block,
so L that (1) holds and Lv is an A5-block. Further as Lv is trivial on R/Rv, V,,
is the unique non-central chief factor for Lv on R, so Vv is the unique noncentral
chief factor for Lv on 02(KvS). Thus Kv is also a block. By 12.6.15, U is an FF-
module for KZS*, so by Theorem B.4.2, K; is either of Lie type and characteristic
2, or A1. (The case of A 6 is ruled out as Lv/02(Lv) ~ A 6 ). Indeed as SL 3 (2n)and G2(2n) have no subgroup X with X/02(X) ~ A5, KZ ~ Sp4(2n), A1, As, or
L5(2). As T; acts on L~ and IS* : T;I :::; 4 = IT : Tvl, KZ is A1, As, or L5(2).
Furthermore in the latter two cases IS: Tvl = 4 = IT: Tvl, so that SE Syl2(G),
and we calculate that R = Rv, and L~ ~ A 6 or A5/ E 16 , respectively. In particular
KZ is not L 5 (2), since in that group [0 2 (L~), R*] # 1, whereas Lv has a unique
noncentral 2-chief factor. Then as V,, :::; [U, Kv], Theorem B.5.1.1 says U /Cu ( Kv)
is the natural module for KZ ~ A1 or As. Now we argue as in the proof of 12.6.17:
In either case [v 0 , Kv] -=j:. 1, so as v E Z(Gv), v # v 0 and hence (3) holds. Finally