876 13. MID-SIZE GROUPS OVER F2
Next assume W 1 (T, U) does not centralize U. Then by the previous paragraph,
there is g E G with A:= UYnT a hyperplane of UY and A =J. 1. As A =J. l, I.6.2.2a
says that A* is the full group ofF 2 -transvections on U with axis UnMg. Inspecting
the cases listed in 13.1.10, we conclude L ~ £3(2), m(U) = 3, and E4 ~A::::; L.
Hence A induces a faithful 4-group of inner automorphisms on L/OinJy(L). This
is impossible as L/02(L) ~ SL2(7)/E49, so Aut(L/02(L)) ~ GL2(7)/E49. This
completes the proof of (1).
By (1), w(G, U) > 1, and by 13.1.13, r(G, U) = m(U) > 1. Thus (2) holds.
Finally the hypotheses of E.3.35 are satisfied with U, R, Mc in the roles of "V, Q,
M", so (2) and E.3.35.1 imply (3). D
We are now in a position to complete the proof of Theorem 13.1.7.We saw at that outset of the proof that IM(T)I > 1, so that there is some
M E M(T) with M =J. Mc. ·Thus we may choose Y as in 13.1.6. Then Y i Mc,
so n(YT) > 1 by 13.1.14.3. Thus Y is not solvable by E.1.13, so case (2) of 13.1.6
holds, and in particular YE .Cj(G, T). Therefore Y/0 2 (Y) is described in 13.1.2.3,
so as n(YT) > 1, Y/02(Y) ~ A5 using E.1.14.
Next as Y i Mc, Y i Na(Wo(T, U)) in view of 13.1.14.1. Thus by E.3.15,
there is A:= UY::::; T for some g E G with Ai Q := 02 (YT). Let AQ :=An Q;
then m(A/AQ)::::; m 2 (YT/0 2 (YT)) = 2, so as m(A)?:: 3 by 13.1.10, it follows that
AQ =J. 1.
As Ai 02(YT), 02 ((AY)) = Y. As Yi Mc, there is h E Y with Ah i Mc.
But Ah ::::; YT, so if U::::; 02 (YT) = Q, then (Ah, U) is a 2-group with 1 =J. A~ =
Ah n Q ::::; Na(U); then by I.7.6, Ah ::::; Ca(U) ::::; Na(U) = Mc, contrary to our
choice of Ah. Thus U i Q, so we may take A= U.
Let I := (U, Uh). Then as m 2 (YT/Q) ::::; 2 and Q = keryT(NYT(U)), V :=
Un M~ and B := Uh n Mc are of codimension at most 2 in U and Uh, respectively.
Therefore since I/0 2 (I) is a section of Y/0 2 (Y) ~ A 5 , (a) and (c) of I.6.2.2 say
that 02(I) = V x B, and I/02(I) ~ D5, D10, or L2(4), and 02(I) is a direct sum
of natural modules for I/0 2 (I). However in the first two cases, Bis of index 2
in Uh, so by 13.1.14.1, [B, U] = 1, and then [Uh, U] = 1, a contradiction. Hence
I/02(I) ~ L2(4). Let DE Syl3(N1(U)); then V = [V, D] since 02(J) is the sume
of natural modules for I/0 2 (I), and so U = [U,D]; thus U is the natural module
for L* ~ L2(4) by 13.1.10. Hence B ~ E4 and V = Cu(b) for each b EE#, so B
induces a faithful 4-group of inner automorphisms on L/0 00 (£). ·As in the proof of
13.1.14, this is a contradiction. This completes the proof of Theorem 13.1.7.
13.2. Some preliminary results on A 5 and A 6
In this section we establish some technical results used in our treatment of
the cases L/02,z(L) ~ A5 or A6 in the FSU. Thus in section 13.2, we assume
Hypothesis 12.2.3 from the previous chapter. In particular M = Na(L) for some
LE .Cj(G, T) with L/02(£) quasisimple and VE Irr +(L, R 2 (LT)) is T-invariant
and satisfies the Fundamental Setup (3.2.1).
As usual we adopt the conventions of Notation 12.2.5; e.g., Z = fh(Z(T)),
Mv = Na(V), and Mv = Mv /Ca(V). We also set
Zv := Cv(L) and V := V/Zv.
Throughout this short section we assume that L ~ An for n = 5 or 6. Then
we are in case (d) of 12.2.2.3, with V the 4-dimensional chief factor in a rank-n