i3.7. FINISHING THE TREATMENT OF Aa WHEN (vG1) IS NONABELIAN 939(a) n = 6, L/02(L) ~ A5, V2 = (ei,2,3,4), and Li has two noncentral chief
factors on UH, or
(b) n = 7, L/02(L) ~ A.6, V2 = (e5,5), and V3 = (e5,5, e5,7)·
(4) If K/02(K) ~ A1 then UK is not a 4-dimensional A 7 -module.
(5) If K/02(K) ~ (8)L3(2n), 8p4(2n)', or G2(2n)' and UK/Cu-K(K) is a
natural module for K*, then n = 1.PROOF. Assume K/02(K), UK is one of the pairs considered in the lemma. We
obtain a contradiction in (2) and (4), and in (5) under the assumption that n > 1.
In (1) and (3), we establish the indicated restrictions. Observe that, except possibly
in (5) when K/02(K) ~ 8L3(2n), K/0 2 (K) is simple so that K* ~ K/0 2 (K). In
that exceptional case UK is a natural module by hypothesis, so CK(UK) = 02 (K)
and thus again K* ~ K / 02 ( K).
The first part of the proof treats the case where Li :::; K. Here K ~ H by
13.7.5.1, and UH= UK by 13.7.5.2.
Next V 3 = (V 2 £^1 ) is a T-invariant line in UK, so:
(i) If K* ~ L3(2) then Cu-H(K) = 0 (cf. B.4.8.2).
(ii) Under the hypotheses of (3), n > 5.Further
(iii) V 2 is a T-invariant F 2 -point of UH. Set Ko := 02 (CK(Vi)), so that also
Ko= 02 (CK(V2)).
By 13.5.4.3, m 3 (Ko) :::; 1, so we conclude from (iii) and the structure of CK* (V2)
that: (2) holds; n < 5 in (1); in (3), n:::; 7 and in case of equality V2 = (e5,5), when
UH is described in the notation of section B.3.
Assume the hypothesis of (3) with n = 7. As Vi - = (e - -L
5 ,6) and Vi = (112^1 ) is a
T-invariant line, V3 = (e5,5, e5,7)· Hence NK(Vi) has 3-rank 2, so L/02(L) ~ A.6
by 13.7.5.4. Since NGL(UK)(K*) ~ 87, H =KT. Hence conclusion (b) of (3) holds.
Thus under the hypotheses of (3), we have reduced to the case n = 6. Then as
V 3 = (V 2 £^1 ) is a T-invariant line, V2 = (ei,2,3,4), Li has tw!=J noncentral chief factors
on UH, and m3(NK(V3)) = 1, so that L/02(L) ~ A5 by 13.7.5.5. Since EndK(UK)
is of order 2, we conclude that K = F(H*). As T normalizes UK, it is trivial on
the Dynkin diagram of K, so as Out(K) is a 2-group, we conclude that H =KT.
This gives conclusion (a) of (3), and so completes the proof of (3).
Similarly when n = 4 in case (1), or in case (4), NK(Vi) has 3-rank 2, so
that L/02(L) ~ A6 and Li = 02 (NK(Vi)) by 13.7.5.4. Thus LiT/02(LiT) ~
83 x Z 3 or 83 x 83 from the structure of L. Further as UH = UK and K ~ H,
H = NaL(UH)(K) = K. Thus in case (4) where K ~ A1, LiT/02(LiT) is
E 9 extended by an involution inverting the E9, so this case is eliminated. When
K* ~ L 4 (2), the conclusions of (1) hold using I.1.6.6.
Assume the hypotheses of (5) with n > 1, and let U"jj := UH/CuH(K). By
(iii), v 2 + is contained in a T-invariant F2n-point W of U"jj. As Li :S K and Li
is T-invariant, Li is contained in the Borel subgroup of K containing T n K. In
particular, n is even. Thus Li acts on W, so v 3 + = [V 2 +, Li] :S W. But now
as 031 (CK(W)) is not solvable, 13.7.5.6 supplies a contradiction, establishing that
n = 1 under the hypotheses of (5).