12.7. THE TREATMENT OF Aa ON A 6-DIMENSIONAL MODULE 83912.2.13 until after both He and M 2 4 have been independently identified; 12.2.13 is
only used when we are working toward the final contradiction.
We mention that the groups He and M 24 will be identified via Theorem 44.4
in [Asc94] in our Background References.12.7.1. Preliminary results. The proof of Theorem 12.7.1 involves a series
of reductions.
Assume L, V arise in a counterexample G. Then by Theorem 12.2.2, V is
a 6-dimensional module for L/CL(V) ~ A 6 and CL(V) = 02 (L). We adopt the
conventions of Notation 12.2.5. Let TL:= T n L, and P 1 and P 2 the two maximal
subgroups of LQ containing TLQ. Let Ri := 02 (Pi) and X := 02 (0 2 ,z(L)). We
can regard V as a 3-dimensional vector space F V over F := F 4 , with L :::; SL(F V)
and X inducing F-scalars on F V.
LEMMA 12.7.2. (1) Either Mv = L ~ A 6 or Mv = Lf' ~ S5.
(2) P1T is the stabilizer in LT of an F-point Vi of F V, and R1 ~ E 4 is a group
of F-transvections on pV with center V1 and Cv(R1) = V 1.
(3) P1 is irreducible on V/V 1 and V1.(4) P2T is the stabilizer of an F-line Vz of pV, and R2 ~ E4 is a group of
F -transvections on F V with axis Vz and [V, R2] = Vz.
(5) 02 (Pi) = LiX, where {Li,X} are the unique T-invariant subgroups I=02 (1) of Pi with II: 02(I)I = 3.
(6) L = e(M) is the characteristic subgroup of M generated by all elelements
of order 3 in M.
PROOF. The calculations in (1)-(5) are well-known and easy. Notice in (1)that automorphisms of A 6 = Sp4(2)^1 nontrivial on the Dynkin diagram are ruled
out, as they do not preserve V. Part (6) follows from 12.2.8. D
In the remainder of the section, we adopt the notation of Li and L2 as in
12.7.2.5.
LEMMA 12.7.3. A2(f', V) = {R2} and a(Mv, V) = 2.
PROOF. Let A E A 2 (f', V). Then Cv(A) = Cv(B) for each hyperplane B of
A, so as 1 I= A, m(A) > 1. If A f:. L, then B := An L is a hyperplane of A, and
Cv(B) is an F-subspace of V, whereas a EA - Lis nontrival on each a-invariant
F-point since a inverts X. We conclude that A :::; L, so as Ri, i = 1, 2, are ofrank 2 and are the maximal elementary abelian subgroups of TL, A = Ri for some
i. By 12.7.2.2, i I= l, and 12.7.2.4 shows that R2 E A2(f', V), so A = R2. Since
m(R2) = 2, a(Mv, V) = 2. D
LEMMA 12.7.4. (1) L has two orbits on V# with representatives z E Vi and t.
(2) Let vt be the F-point of pV containing t. Then NLCV't) ~ GL2(4), and Vis an indecompos.able module for CL(vt) with V/vt the natural module.
(3) t EV= [V,Lt] :S Lt and Lt= e(Mt)·
(4) Vz is partitioned by two conjugates of vt and three conjugates of V1.
PROOF. From 12.7.2, IVPI =IL: P1I = 15, leaving a set 0 of 6 F-points of pV
not in V 1 L. As 6 is the minimal degree of a faithful permutation representation for
L/ X ~ A 6 , it follows that Lis transitive on 0 (so that (1) holds), and the stabilizer
in L of vt E 0 is isomorphic to G L 2 ( 4). As V = [V, X], V /vt is the natural
module for NL(vt) ~ GL2(4), so a Sylow 2-subgroup S of Nt(vt) centralizes an