4.4.1. Remark. In order to construct an appropriate subgroup X as above, we assume that the subgroup A of S is G-completely reducible. For this, we appeal to Theorem 4, hence Theorem 1 depends on Theorem 4 in these few cases. We take this opportunity to note that the proof of Theorem 4 given in Chapter 5 depends only on the feasible characters of simple subgroups (Theorem 2), and not on Theorem 1.
Case: G = E7, H = Alt10 or Alt9, p 6= 2, 3. Here the composition factors
of S on L(G) and Vmin are specified by one of the tables on pages 92-93. Let
A ∼= Alt8 be a subgroup of S, lying in an intermediate subgroup isomorphic to
Alt9. Then this subgroup Alt9 has an 8-dimensional composition factor on L(G),
whose restriction to A contains a trivial submodule. Thus Proposition 3.6 applies to L(G) ↓ A, and A is not Lie primitive in G. Since p 6= 2, 3, Theorem 4 holds and A is G-completely reducible. In addition, by Theorem 0 there does not exist an embedding of Alt8 into a smaller exceptional algebraic group, and thus A lies
in a G-completely reducible semisimple subgroup, whose simple factors are all of classical type.
Now Alt8 has a 7-dimensional irreducible module, giving an embedding into
B3, and no other non-trivial irreducible modules dimension ≤ 12. In addition, the
only non-trivial faithful module for 2 · Alt8 of dimension ≤ 12 is 8-dimensional,
giving an embedding into D4. If 2 · Alt8 embeds into a simply connected group of
type D4 with its centre contained in Z(D4), then its centre acts trivially on one
of the three D4-modules λ1, λ3 or λ4. Since these are self-dual, this restricts as a
direct-sum 1 ⊕ 7, and therefore the quotient Alt8 in D4 lies in a proper subgroup
of type B3.
Thus A lies in a subgroup B3of G. The two conjugacy classes of such subgroups
and their action on VG(λ7) are given by [55, Table 8.6]. Comparing these with the
feasible characters of A (Tables 6.123, 6.122, 6.121), we see that A lies in a subgroup X = B3 < A6 with VG(λ7) ↓ X = λ21/λ22. By Lemma 2.10 and Proposition 2.7,
this is completely reducible. Thus Proposition 4.7 applies and every A-submodule of VG(λ7) is X-stable. In particular, every S-submodule of VG(λ7) is preserved by
Y = hS, Xi, hence if M denotes the (non-empty) collection of proper S-submodules of VG(λ7), we have S < Y < TM∈MGM
◦
, and this latter group is proper, connected and NAut(G)(S)-stable.
Case: (G, H, p) = (E8, Alt16, 2). Let A ∼= Alt15 be a subgroup of S. Then
Proposition 3.12 and Theorem 4 apply to A, and A is not Lie primitive in G, and is G-cr. Since the smallest nontrivial A-module is 14-dimensional and H1(A, 14) = 0,
and since A does not embed into a smaller exceptional algebraic group, we deduce that A lies in a Levi subgroup of type D7, call it X. From [55, Table 8.1] it follows
that
L(G) ↓ X = (02/VX(λ2)) ⊕ λ21⊕ λ6⊕ λ7.
Comparing this with L(G) ↓ A, whose factors are given by Table 6.235, we find that A must be irreducible on each X-composition factor. In particular every 14- and 64-dimensional A-submodule of L(G) is the restriction of an X-submodule.
Now, if S has no irreducible submodules of dimension 1, 14 or 64 on L(G), then L(G) ↓ S is an image of the projective cover of the 90-dimensional S-module. But
this is absurd, since this composition factor is self-dual and occurs with multiplicity 1. Thus either S has a nonzero trivial submodule on L(G) or has irreducible submodules of dimension 14 or 64, in which case hS, Xi◦ preserves every such submodule, and is therefore a proper, connected subgroup containing S.
So S is not Lie primitive in G; hence by Lemma 4.2 we may assume that S is G-cr. Since the smallest S-module is 14-dimensional and supports a quadratic form, we deduce that S lies in a subgroup D7; this is the subgroup X constructed
above, and we deduce that every 14- and 64-dimensional S-composition factor on L(G) is in fact an X-submodule. Thus S < X < (T
MGM)◦, the intersection over
14- and 64-dimensional S-submodules, and by Corollary 4.5 this latter group is proper, connected and NAut(G)(S)-stable.
Case: (G, H, p) = (E8, Alt11, 11) or (E8, Alt10, p 6= 2, 3, 5). Assume first that
p 6= 7. Note that each feasible character of Alt11 or Alt10 on L(G) has two 84-
dimensional composition factors (Table 6.235). Let S ∼= H be a subgroup of G, and let A ∼= Alt9be a subgroup of S. Matching up Tables 6.235 and 6.238, we see that
L(G) ↓ A = 1/83/27/283/562,
which is completely reducible since p ∤ |A|. Thus A fixes a vector on L(G) and lies in a proper subgroup of positive dimension, hence also in a proper, connected subgroup by Proposition 3.11. By Theorem 4, A also is G-completely reducible.
Inspecting [41, Table 2] we see that the smallest irreducible modules for A and its proper cover 2.A are of dimension 8 and 21, and the 8-dimensional modules give an embedding into D4. Since Alt9admits no embeddings into a proper excep-
tional subgroup of G by Theorem 0, it follows that a minimal connected reductive subgroup containing A can only involve factors of type D4. Comparing the above
decomposition with [55, Table 8.1], we deduce that A lies in a simple subgroup X of type D4, contained in a Levi subgroup A7. Then
L(G) ↓ X = 0/λ21/λ32/2λ1/(λ3+ λ4)2,
which is completely reducible by Corollary 2.12 and Proposition 2.7. Now, let W be the span of all 28-dimensional and 56-dimensional A-submodules of L(G). Then W is the restriction of an X-submodule with factors λ3
2/(λ3+ λ4)2, which contains
every 28- and 56-dimensional X-composition factor of L(G). Proposition 4.7 then applies, so that every A-submodule of W is X-invariant.
Now, as L(G) is self-dual, S has either a unique 84-dimensional submodule or has a summand of shape 84 + 84. In either case, this is contained in W , since ‘84’ must restrict to A with composition factors 28/56. Thus each 84-dimensional irreducible S-submodule of L(G) is an X-submodule, hence preserved by Y def= hS, Xi. Thus S < Y ≤ (T GM)◦, the intersection over 84-dimensional irreducible
S-submodules of L(G). This latter group is NAut(G)(S)-stable by Corollary 4.5.
If p = 7, then the same argument goes through with the caveat that L(G) ↓ A may no longer be completely reducible; however, it still has a trivial submodule since H1(Alt
9, M ) vanishes for each composition factor M in a feasible character
(cf. Table A.2). The 28- and 56-dimensional A-modules are projective, since their dimension is divisible by the order of a Sylow subgroup of A, and therefore they are A-direct summands of L(G).