752 10. THE CASE LE Cj(G, T) NOT NORMAL IN M.A 7 -block, or there is a nontrivial characteristic subgroup C of 8 normal in ITv.However in the last case G 0 := (I, T) ::::; Na(C), so as L ::::; I, LoT ::::; Go and
hence I::::; Go ::::; M = !M(L 0 T). This contradicts Ii M, so I is a block. Further
if I is an A 7 -block, then as I = [I, J(Tv)], I*T;: ~ 87, so L/02(L) is not L3(2)
as LE .C(ITv,Tv)· If I is an A1-block, then I* is self-normalizing in GL(Vr), so
I*T;: =I*. Thus (bl) or (b2) holds.
In particular in case (b), 02 (!) = Cr(Vr). In case (c) since I/02(I) is quasisim-
ple, the list of Schur multipliers· in I.1.3 says I/ 02 (I) ~ I*, so again 02 (I) = C 1 (V1).
Assume L ~ L 3 (2); this argument will be fairly lengthy. By 10.2.7, case (3)or (5) of 10.l.l holds. In case (b ), subcase (bl) holds; so L* is self-normalizing in
I*T;: ~ A1, and hence Tv induces inner automorphisms on L so that case (3) of
10.l.l holds. Similarly in case (c): if I~ L 4 (2), then L ~ L3(2)/Es, and so Tv
induces inner automorphisms on L and L is self-normalizing in I; while if I* ~
Ls(2), then either Tv induces inner automorphisms on L, or I*T;: ~ Aut(Ls(2)),
L* is the Tv-invariant nonsolvable rank-2 parabolic, and L* is self-normalizing inI*. Except in this last case, case (3) of 10.l.l holds.
Set Y := 02 (CL 2 (v)). In case (3) of 10.1.1, Y/02(Y) ~ Z3. In case (5) of
10.1.1, either Y/0 2 (Y) ~ Z 3 , or vis diagonally embedded in the two summands
with Y = 1, and Tv = T1 with LTv/02(LTv) ~ Aut(L3(2)).
Suppose Y-!-l. By A.3.18, I= 031 (H) so Y::::; N1(L). As we saw C1(V1) =02 (!), 1-!-Y ::::; N1• (L) and Y i L. Thus L < 02 (N 1 • (L), so by the previous
two paragraphs, I*T;: ~ L 5 (2), Y* L*T;: ~ 83 x L 3 (2), and case (3) of 10.l.1 holds.On the other hand ifY = 1, then by the previous two paragraphs, case (5) of 10.l.l
holds, and IT;: ~ Aut(L 5 (2)). Therefore in any case for Y, I~ L 5 (2).
Suppose that Cv(L 0 ) -!-l. Then case (3) of 10.l.1 holds by 10.1.2.4, so by the
previous paragraph, LYTv/02(LYTv) ~ L2(2) x L3(2), contrary to 10.2.8.4, which
says that LYTv/02(LYTv) ~ Z3 x L3(2).
Therefore Cv(L 0 ) = l. By B.4.2 and Theorem B.5.1 Vr is either an irreducible
of rank either 5 or 10, the sum of the 5-dimensional module and its dual, or the sum
of isomorphic 5-dimensional modules. If Y = 1, we saw that I*T;: ~ Aut(L 5 (2))and L* is the nonsolvable T;:-invariant rank 2 parabolic. Thus V1 ·= Vr,1 EB Vr,2
with Vr,1 a natural !*-submodule and Vr,2 its dual. But we also saw that case (5)
of 10.l.1 holds, and in that case we saw that Vv = Vi ::::; V[. However Vi is the
sum of a natural module for L and its dual, whereas the parabolic L * has no such
submodule on V1.Thus Y-!-1, IT;: ~ Ls(2), and LY*T;: ~ 83 x L 3 (2). In case (5) of 10.1.1,
V1 ::::; Vr and Vi is the sum of a natural module for L and its dual. However
examining the possibilities for Vr listed above, we see that the parabolic L YT;:
has no such submodule.Therefore case (3) of 10.l.1 holds. Since Cv(L 0 ) = 1, Vi is the natural module
for L. But from the our list of possibilities for V1, each natural submodule for L is
contained in an I-irreducible. Thus as VI= (Vl), Vr is an I-irreducible, and hence
dim(Vr) = 5 or 10.
Again since Cv (Lo) = 1, Tv = T1, so that T normalizes Tv. Let t E T - Tv,
u := vt, and E := (u, v). '!'hen (u) = Cv 1 (Tv) and CaJE) = CaJu). Since V1 isan irreducible of dimension 5 or 10, C1·r;;(u) is a maximal parabolic of I*T;:, and
so from the structure of such parabolics,C1rv(E) = 0
31
(Ca(E))Tv::::; ItTv,