13.5. THE TREATMENT OF A5 AND Aa WHEN (v;^1 ) IS NONABELIAN 925
LEMMA 13.5.23. K* is not As.
PROOF. Assume K* ~As. Then by 13.5.21.3 and I.l.6.1, UH is either the 7-
dimensional core of the permutation module for K*, or its 6-dimensional irreducible
quotient, which we regard as an orthogonal space for K* ~ nt(2). By 13.5.21.5,
[UH, V*] = V3 is of rank 2, while if UH were 7-dimensional, then [UH, V*] would be
of rank 3. Therefore UH is 6-dimensional orthogonal space. Moreover CK(V 2 )* is
of 3-rank 2, so n = 5 by 13.5.4.3.By 13.5.21.5, UH = [QH, K], and LiT is the minimal parabolic of KT*
stabilizing the 2-space V 2. Thus £ 1 has exactly four noncentral 2-chief factors, two
on QH and two on 02(L1)* = 02(Li) ~ Q~. Therefore by 13.5.22.2, the parameter
s of 13.5.22 is equal to l. Thus by 13.5.22.3,
JUHJ = 22 s+5JS2: VJ= 27 · JS2: VJ. (*)
Next as UH is the orthogonal module, UH ~ D~ is of order 27. Thus 82 = Y
by (*), and ISi = 2s by 13.5.22.1, soI02(L1)I:::; I02(L1)l · JSI = 2^2. 2s = 210. (**)Further I02(L1)*I = 2^5 using 13.5.21.5, and UH :::; 02 (£ 1 ) by 13.5.22.4, so that
J0 2 (L 1 )J 2 212 by(*), contrary to(**). This contradiction completes the proof of
13.5.23. D
LEMMA 13.5.24. K* is not A5.
PROOF. Assume K* ~ A5. Then by 13.5.21.3 and I.l.6.1, UH is either the 5-
dimensional core of the permutation module for K*, or its 4-dimensional irreducible
quotient. In either case by 13.5.21.5, UH = [QH, K] and LiT* is the maximal
parabolic stabilizing the line V3. Thus L1 has two noncentral chief factors on UH,
and one on 02 (Li), so L1 has exactly three noncentral 2-chief factors. This is a
contradiction, since by 13.5.22.2, the number of noncentral 2-chief factors of £ 1 is
even. This completes the proof of 13.5.24. D
LEMMA 13.5.25. K* is not G2(2)'.
PROOF. Assume K* ~ G 2 (2)'. Then by 13.5.21.3 and I.l.6.5, UH is either
the 7-dimensional Weyl module for K* or its 6-dimensional irreducible quotient.
However UH= Z(UH)Uo where U 0 is extraspecial, and if (Z(UH))·= 1, then H
preserves a quadratic from on UH/Z(U). Therefore as G2(2)' does not preserve a
quadratic form on its 6-dimensional module, we conclude m(U) = 7 and Z(U H) =
(j) ~ Z4.
Let XE Syh(L1); then (j) = CuH(X). But by 13.5.22.4, UH:::; 02(L1), and
by G.2.4.6, S/82 is the sum of natural modules for L_/8. So j E Co 2 (Li)(X):::; 82.
However 82 = V(UH n U_k) with (UH n U_k) :::; (UH) n (U_k) = V1 n Vf = 1, so
V1 = ( (j)) :::; (82) = (V)(U H n u_k) = 1,
a contradiction. D
By 13.5.21.2, K* is A 6 , As, or G 2 (2)'. But this contradicts 13.5.24, 13.5.23,
and 13.5.25. This contradiction completes the proof of Theorem 13.5.12.