13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 907holds by A.1.7.2. Finally consider any e E Ov(d) - {d}. Then e E Oa(d)::::; E, so
since E controls fusion of its involutions, by conjugating in E we may assume that
e E fJi. Then de E "(2, so that Oa(de) ::::; Eby the previous paragraph, verifying
condition ( c) of the definition and establishing the claim.
Therefore as G is simple, we may apply Corollary ZD on page 22 of [GLS99],to conclude that G is a simple Bender group, and E is a Borel subgroup, which is
strongly embedded in G. This is impossible by 7.6 in [Asc94], as E has more than
one class of involutions. D13.4.3. Eliminating the case Gz nonsolvable. If Gz is solvable then The-
orem 13.4.1 holds by Proposition 13.4.9. Thus we may assume for the remainder
of the proof of the Theorem that Gz is not solvable, and we will work to a contra-
diction.In particular there exist nonsolvable members of 1iz. Our first result is a
refinement of the information produced earlier in 13.4.6.
LEMMA 13.4.10. Let HE 1iz be nonsolvable. Then
{1) There exists K E C(H), K i. M, K E .Cj(G, T), and K <l H. SetVH := (ZK) and (KT):= KT/OKr(VH); then K ~ £3(2) or A5.
{2) L1::::; K.
{3) L/0 2 (L) ~ A5 and if K* ~ A5, then K/02(K) ~ A5.
(4) Let VK := [VH, K]. Then VK = [R2 (KT), K] and either VK is the natural
module for K*, or VK is a 5-dimensional module for K* ~ A6 with (z) = OvK (K).
(5) IZI = 4, so Z =Zn V =Vi, IZLI = 2, and Oz(K) = (z).
{6) Zv = ZL.
(7) M =LT and H =KT= Gz. Thus 1iz = {Gz} and VH = (ZH). Further-more Gz contains a unique member of 1i*(T, M): the minimal parabolic of H over
T distinct from L1T.
{8) Let H 2 E 1i(T) be the minimal parabolic of H distinct from L1T, and set
Ho := (H 2 , L 2 T). Then Ho E 1i(T), H2 is the unique member of 1i* (T, M) in Ho,
and either:{i) Conclusion {i) of 13.4. 7.1 holds, z is of weight 4 in V, and Zv::::; VK.
Further if VK is a 5-dimensional module for K* ~ A5, then Zv is of weight 4 in
VK·
(ii) Conclusion {ii) of 13.4. 7.1 holds and Ho= Na(J(T)) E M(T).PROOF. First by 13.4.6.1, there exists KE C(H), KE .Cj(G, T), K :":! H, and
Ki. M. By 13.4.5.1, K/02(K) is A5, £3(2), A5, or A5.
Set U := [R 2 (KT), K]. By 13.4.5.3 with KT in the role of "H", there is
WK E Irr+(K,R 2 (KT),T), and for each such WK, WK = ((Zn WK)K) ::::; U
and WK is either a natural module for K/0 2 ,z(K) or a 5-dimensional module forK/02,z(K) ~ A5.
Now K = [K, J(T)] by 13.4.6.3, so Theorem B.5.1 shows that either U E
Irr +(K, R 2 (KT)), or U is the sum of two isomorphic natural modules for K* ~£ 3 (2), which are T-invariant since then T ::::; K. In particular U is the A5-module
if K* ~ A 5 , and U =((Zn U)K)::::; VH, so U = VK. By B.2.14, VH = UCz(K), so
CKr(U) = CKr(VH ).
As VH = UOz(K), OKr(Z) = OK(Z n U)T, so that OK(Z)*T* is a max-