6.2. IDENTIFYING M22 VIA £ 2 (4) ON THE NATURAL MODULE 69iPROOF. By 6.2.13, we may assume Zs < Un V, so the subsequent lemmas in
this section are applicable. In particular by 6.2.18.2, H and its action on f) are
described in Theorem G.11.2.
By 6.2.16.4, V is generated by an involution v of type a 2 in 8p(U) and by
6.2.17.9, v E X. However in cases (8) and (10)-(13) of G.11.2, X contains no
involution i with m([U, i]) = 2, so none of these cases holds. Similarly in case (9),
we must have H =Hi x H2 with H 9:! 85, H2 9:! L2(2), U is the tensor product of
the natural modules for Hi and H2, and vis a transposition in Hi. But then H 2 is
transitive on [U,v]#, contrary to parts (4) and (6) of 6.2.16. The same argument
eliminates case (3) of G.11.2, as there v centralizes Z(O(H)) which is transitive on[U,v] A #.
Let d := dim(U). By 6.2.15.1, d ::'.". 4, so case (1) of Theorem G.11.2 does not
hold.
In case (2) of G.11.2, d = 4 so 8p(U) 9:! 86 acts naturally on U. Thus as v is oftype a2, vis of cycle type 23 in 86 and 3 E 7r(H), so 15 or 18 divides IHI by G.11.2.
Therefore His 86, 85 with fj the L2(4)-module, or a subgroup of ot(2) of order
divisible by 9. In each case NH(F) is transitive on ft# for each totally singular line
F in U, contrary to 6.2.16.6.
As v is of type a 2 in 8pa(2), [vvhl ::; 4 for each h E H. Thus in case (4) of
Theorem G.11.2, v is a transposition; in case (5), v is a transposition or of type 24 ;
in case (6), vis a long root involution; and case (7) is eliminated. As m([U, v]) =Vu
is of rank 2, while transpositions in cases ( 4) and (5) act as transvections on fJ, we
conclude that case (4) does not hold, and in case (5), that vis of type 24. But nowN fl(Vu) is transitive on V-;lj, contrary to 6.2.16.6. This contradiction completes the
proof of the Theorem. DWe summarize the work of the previous two chapters in:
THEOREM 6.2.20. Assume G is a simple QTKE-group, T E 8yl2(G), L E
Lf(G, T) with L/02(L) 9:! L2(2n) and L :sJ ME M(T), and V E 'R'2(LT) with
[V, L] f. l. Then one of the following holds: ·
(1) L/0 2 (L) 9:! A5, and [V, L] is the sum of at most two A5-modules for
L/0 2 (L). Further n(H) = 1 for all HE 1-i*(T, M).
(2) G is a rank-2 group of Lie type and characteristic 2, but G is U5(q) only if
q= 4.
(3) G 9:! M22 or M2s-PR00F. Suppose first that Hypothesis 5.1.8 holds. Then we may apply The-
orem 5.2.3, whose conclusions are among those of (2) and (3) in Theorem 6.2.20.Thus we may suppose that Hypothesis 5.1.8 fails, and hence n(H) = 1 for all
HE 1-i*(T, M). Then we are done if the first statement in conclusion (1) of 6.2.20
holds; so we may assume it fails, and then we have Hypothesis 6.1.1. Then Theorem6.2.19 says G 9:! M 22 , so that (3) holds. D
In particular, since the groups in conclusions (2) and (3) appear in the list of
our Main Theorem, the treatment of QTKE-groups G containing some T-invariant