13.3. STARTING MID-SIZED GROUPS OVER F 2 , AND ELIMINATING U 3 (3) 885( 3) There is a T -invariant V K E Irr+ ( K, R2 (KT)) and further each member of
Irr+(K,R2(KT),T) is T-invariant. The pair K, VK satisfies the FSU and either
VK is the n·atural module for K/CK(VK) ~ A 5 , A 6 , L 3 (2), or U 3 (3), or VK is the
5-dimensional core of a 6-dimensional permutation module for K/CK(VK) ~ A 6.
(4) Hypotheses 13.1.1, 12.2.1, and 12.2.3 are satisfied with K in the role of
"L".
(5) Hypothesis 13.3.1 is satisfied with Kin the role of "L" unless K/0 2 (K) is
A5 but L/02(L) is not A5.
PROOF. The initial argument is similar to that in 13.1.2: First K s I E
£*(G, T), and by 1.2.9, IE £j(G, T). By Theorem 13.1.7, I/0 2 (!) is quasisimple,
so K = I by 13.1.2.5. Therefore Hypothesis 12.2.3 holds with Kin the role of "L" by
13.1.2.1. Hence (4) is established. Furthermore parts (1)-(3) of Hypothesis 13.3.1
are satisfied by Kin the role of "L', so (5) also follows as part (4) of Hypothesis
13.3.1.4 is satisfied by K unless K/0 2 (K) is A 5 , but L/0 2 (L) is not.Part (1) follows from 13.1.2.3. Further 13.1.2 says that K is T-invariant and
the first sentence of (3) holds. Then Na(K) = !M(KT) by 1.2.7.3, completing the
proof of (2).It remains to show VK is one of the modules described in (3). Theorem 12.2.2.3
supplies an initial list of possibilities for VK, and by Remark 12.2.4 the list of
12.2.2.3 can be refined using results from the previous chapter. If CvK (K) =I-1,
then Theorem 12.4.2 rules out the indecomposables in cases (b) and (f) of 12.2.2.3,
leaving only case (d) with VK the core of a 6-dimensional permutation module for
K/CK(VK) ~ A 6 • Otherwise CvK(K) = 1, so either VK is one of the natural
modules listed in 13.3.2.3, or VK is the 6-dimensional faithful module for A5. The
last case is out by Theorem 12.7.1 and the exclusions in Hypothesis 13.3.1.3. D
Of course we may apply 13.3.2 to Lin the role of "K", so VE Irr +(L, R 2 (LT))
is T-invariant and V is one of the modules listed in 13.3.2.3. By 13.3.2, L satisfies
Hypothesis 12.2.3, so we may appeal to the results from the previous chapter, and
when L /CL (V) ~ A 5 or A 6 we may appeal to results from section 13.2 of this
chapter. We adopt the conventions in Notation 12.2.5 from the previous chapter.
We will refer to a module V which is the core of a 6-dimensional permutation
module for L/CL(V) ~ A 6 as a 5-dimensional module for A5. In addition we adopt:
NOTATION 13.3.3. If L ~ L3(2), A5, or A5, define the T-invariant subspaces
Vi of V for 1 S i S dim(V/Cv(L)) as in Notations 12.8.2 and 13.2.1. When L
is U 3 (3), V is the 6-dimensional module for L regarded as G2(2)', which is the
quotient of the Weyl module discussed in [Asc87]; see also B.4.6. In particular, V
admits a symplectic form preserved by Mv, so we can speak of nondegenerate and
totally isotropic subspaces of V. In this case, define Vi to be the unique T-invariant
subspace of V of dimension i. Notice that if Cv(L) = 1, then m(Vi) = i in each
case.
In each case define Gi := Na(Vi), Mi := NM(Vi), and Li := 02 (NL(Vi)).
When L/0 2 (L) is not A5, define Ri := 02(LiT). When L/02(L) ~ A5, define Ri
L 0 , and Li,+ as in Notation 13.2.1.
LEMMA 13.3.4. (1) V1 =Zn V.
(2) V =((Zn V)L).