14.7. FINISHING L 3 (2) WITH (VG1) ABELIANOften we can show that D 7 < U 7 , and in those situations we also adopt:
NOTATION 14.7.1. If D 7 < U,.yi choose"( as in 14.5.18.4, so that
m(U;) 2 m(UH/DH) > 0,1043and u; E Q(H*, UH), and (as in 14.5.18.5) choose h EH with 'Yo = "(2h, and set
a:= "(hand QC<:= 02(Ga); then Ua::::: Ri and u; E Q(H*, UH).
Set Q := 02(LT) = Cr(V) and
s := (Ufi-).
Since Out(L3(2)) is a 2-group and T induces inner automorphisms on L/0 2 (L)
(because T acts on V):14. 7.1. Preliminary reductions.
LEMMA 14.7.2. Let J be a proper H-submodule of Um and assume that Y =
02 (Y) :<::'.'. H with YT/02(YT) ~ 83. Set UH:= UH/I. Then
(1) V is isomorphic to V as an L 1 T-module.(2) (v~n is of rank 1 or 2.
(3) If [V2, Y] = 1, then [V2, Y] = l.
PROOF. Observe as I < UH that V i I since UH = (VH). Then as Li is
irreducible on V, V n I= V 1 , so part (1) follows. Next as V2 is centralized by T of
index 3 in YT, E := (V{) is of rank e = 1, 2, or 3, with E := (V{) of rank e ::::: e.
By Theorem 14.5.3.3, e < 3, so that (2) holds. If [V2, Y] = 1, then e = 1 so E has
the I-dimensional quotient E, and therefore e = 1 or 3. But we just saw e < 3, so
e = 1, and hence (3) holds. D
LEMMA 14.7.3. (1) b 2 3 is odd.
(2) s::::: Q.(3) S is abelian iff b > 3.
(4) If H = G1 and A~:<:::: V for some h EH, then b = 3 and U 7 h E Uj{
PROOF. Part (1) is F.9.11.1. As UH is abelian, UH :5. CLr(V) = Q, so (2)
holds. Part (3) is F.9.14.1, and part (4) follows from F.9.14.3 as Lis transitive on
V# since L = GL(V). D
LEMMA 14.7.4. (1) [V2, 02(G1)] = V1.
(2) I2 := (02(G1)G^2 ):::) G2, I2 = L202(G1), I2/02(I2) ~ 83, and L2 = 02 (I2).
(3) m3(Ca(V2)) :5. 1.
(4) QQH = Ri, so Ri = Q*.PROOF. Part (1) follows from 14.5.21.2, and 13.3.15 implies (2) and (3). By
(1), 1 I=-QH :5. R1, so as Li is irreducible on R1, (4) holds. D
LEMMA 14.7.5. Assume Li :::) H*. Then
(1) D 7 < U 7 , so we may adopt Notation 14.1.1.
(2) QQH = Ri.
(3) Li ~ Z3, and Ri = Q* =Cr• (Li) is of index 2 in T*.
(4) [u;, Li] = 1.