1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

582 3. DETERMINING THE CASES FOR LE .Cj(G, T)

V = VM or Vis a TI-set under M, by 3.2.5. Thus Lo and V are described in 3.2.6,
where V < VM occurs only in subcase (3.c.iii). D

With 3.2.6 in hand, we return to the case in the Fundamental Setup where
L = L 0 , and we obtain more information in the subcase where V = v;,. As in the
proof of the Main Theorem, we divide our analysis into the case where V is an
FF-module and the case where Vis not an FF-module.

LEMMA 3.2.8. Assume the Fundamental Setup {3.2.1) with L =: L 0 and Vo = V.

Assume further that Vis an FF-module for AutaL(V)(L). Then one of the following

~~=.

{1) L ~ L 2 (2n) and V is the natural module.

{2) L ~ SL 3 (2n), and either V is a natural module or V is a 4-dimensional

module for L3(2).

{3) L ~ Sp 4 (2n) and V is a natural module.

(4) L ~ G 2 (2n)' and V is the natural module.

(5) L ~ A5 or A 7 , and V is the natural module.

{6) L ~ A6 and V is a natural module.

{7) Lt ~ A1 and m(V) = 4.

{8) L ~ A.6 and m(V) = 6.

{9) Lt~ Ln(2), n = 4 or 5, and V is a natural module.

{10) L ~ £ 4 (2) and V is the 6-dimensional orthogonal module.

{11) Lt~ L 5 (2) and m(V) = 10.

PROOF. This is a consequence of Theorem B.4.2, using the 1-cohomology of

those modules listed in I.1.6. D

PROPOSITION 3.2.9. Assume the Fundamental Setup FSU {3.2.1), with L =Lo

and Vo= V. Further assume Vis not an FF-module for AutaL(V)(L). Set q :=

q(Lt, V) and q := q(LT, V). Then one of the following holds:

{1) L ~ L2(2^2 n), n > 1, Vis the rr4(2n)-module, and q = q 2:: 3/2, or q 2:: 4/3

if n = 2.

{2) L ~ U 3 (2n), V is a natural module, and q = q = 2.

{3) L ~ Sz(2n), V is a natural module, and q = q = 2.

(4) L ~ (S)L 3 (2^2 n), m(V) = 9n, q > 2, and q = 5/4. Further tis trivial on

the Dynkin diagram of L.

(5) Lt~ Aut(M12), m(V) = 10, q > 2, and q > 1.

{6) L ~ M22, m(V) = 12, and q > 1.

(7) L ~ M22, m(V) = 10, q 2:: 2, q > 1, and q > 2 if V is the cocode module.

{8) L ~ M23, m(V) = 11, q > 2, and q > 1.

{9) L ~ M24, m(V) = 11, q > 2, and q > 1.

PROOF. By hypothesis, Vis not an FF-module for Lt, so J(T) ::::; Cr(V) by

B.2.7; hence we conclude Cv(L) = 0 from 3.2.2.6. Then as VE Irr+(L,R 2 (LT)),

Lis irreducible on V. By 3.2.5, q::::; 2. Then the result follows from the list in B.4.5,

plus the following remarks: The cases in B.4.5 where Lis A 7 or G 2 (2)' do not arise

here because of our hypothesis that Vis not an FF-module for AutaL(V)(L). If

L ~ (S)L 3 (2^2 n) and m(V) = 9n, then V may be regarded as an F 2 n-module, and

F22n ®F 2 n V = N ® Na, where N is the natural F22n-module for SL3(2^2 n) and

a-is the involutory field automorphism of F 2 2n. Hence V is not invariant under an