12.6. ELIMINATING A 8 ON THE PERMUTATION MODULE 823vectors in V, nondegenerate subspaces of V, etc. For i = 1, 2, 5, let Vi denote the
preimage in V of the i-dimensional subspace of V fixed by T. Let Gi := Na(Vi),Mi·== NM(Vi), Li:= 02 (NL(Vi)), and Ri the preimage in T of 02 (L{l').
12.6.1. Preliminary results.
LEMMA 12.6.1. (1) L has two orbits on V#, consisting of the singular and
nonsingular vectors of V.or(2) If Zv =f. 1, then Z2 ~ Zv = Z(T) n V.
(3) Either J(T) = J(Gr(V)), or IR2(LT): VGR 2 (LT)(L)I:::; 2.
(4) J(R1) = J(Gr(V)). Hence Na(R1):::; M.
(5) 0
31(M) = L.
(6) If L = [L, J(T)] and Zv =f. 1, then L centralizes Z.
PROOF. Part (5) follows from 12.2.8. Recall that either
(a) Zv =f. 1, and Vis the 7-dimensional core of the permutation module for L,
(b) Zv = 1, and Vis the 6-dimensional quotient of that core, modulo (en).
Hence (1) and (2) are well known, easy calculations. Also either case (5) or (6) ofB.3.2 holds, so if A E P(T, V), then one of the following holds:
(i) A= (D), for some D ~ ~ = {(1, 2), (3, 4), (5, 6), (7, 8)}.
(ii) A = (~) n L.
(iii) A is conjugate under L to Ao := ( (1, 2) (3, 4), (1, 3)(2, 4), (5, 6), (7, 8)), or to
a hyperplane in the group of (ii), given by either
( (1, 2) (3, 4), (1, 2) (5, 6), (7, 8)), or ( (1, 2)(3, 4), (5, 6), (7, 8)).
(iv) Zv = 1 and A~ E 8 is the unipotent radical of an L 3 (2)/Es parabolic ofL.
Now R 1 ~ E 16 is the unipotent radical of the stabilizer of the partition
{{1,2,3,4},{5,6,7,8}},so R 1 contains no transpositions and hence contains no subgroup of type (i) or
(iii); nor does it contain a subgroup of type (ii) or (iv). Thus R 1 contains no FF*-
offenders, so that J(R1) = J(02(LT)), and hence Na(R1) :::; Na(J(02(LT))) :::;
M = !M(LT), so that (4) holds.
Also if J(T) 1. Gr(V), then V is the unique noncentral chief factor for L on
R 2 (LT) by Theorem B.5.1.1. Then (3) follows as H^1 (L, V) ~ Z 2 by I.1.6.l. If in
addition Zv =f. 1, then Zv = znv by (2), and we've just seen that V = [R 2 (LT), L],so (6) follows as (zL) = VGz(L) by B.2.14. D
In the shadows mentioned earlier (such as Dt(2)), Ga(v) :::; M for each non-
singular v E V, as in the first main reduction Theorem 12.6.2 below. However,
this result does eliminate the sporadic configurations in J4 and F~ 4 , since in those
groups Ga(v) 1. M.
THEOREM 12.6.2. Ga(v) :::; M for each v EV with v nonsingular.
Until the proof of Theorem 12.6.2 is complete, let v E V with v nonsingular
and set Rv := 02 (0Lr(v)). Conjugating in L, we may assume v = ei,2. Thus
Lv ~ A6, so as 02(L) = 02,z(L) = GL(V), Lv = GL(v)^00 , Lv/02(Lv) ~ A5, and