i3.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 903in case (iv), as there Ji(T) :::) Ho. Hence we are in case (ii), so that (4) holds. Thus
we may assume that conclusion (II) of 13.2.2 holds, so that Lis an A 6 -block with
A(02(LT)) <:;;; A(T). As Lis an A5-block, L 2 has exactly three noncentral 2-chief
factors. Let k := 2 in case (ii), and k := 3 in case (iv). As L 2 = K 2 , L 2 has at least k
chief factors on Vo and one on 02(L2)*, so (ii) holds and [0 2 (K 0 T), Ko] :::; V 0. Thus
as J(T) :::) Ho, each A E A(T) contains Vo, so [A,Ko]:::; [0 2 (K 0 T),K 0 ] =Vo:::; A,
and hence A :::l KoA. Further as A(02(LT)) <:;;; A(T), J(0 2 (LT)) =(A E A(T) :
A:::; 02(LT)), so Ko:::; Na(J(02(LT))):::; M, contrary to H 0 i. M by (1).
It remains to prove (3), so we may assume that K 2 = L 2 and conclusion
(iii) holds, and we must produce a contradiction. As m3(L 2 ) = m 3 (K 2 ) = 1
by hypothesis, L/02(L) is A5 rather than A 6 • Let Yz be the preimage in Ho of
Z(0 3 (H 0 )), and set Y2 := 02 (Yz). Notice that as Y 2 * is fixed point free on Vo ofrank
6, while Z =Vi is of rank 2, [Z, Y2] is of rank 4. In particular Y 2 i. Ca(Zv) = M
by 13.4.2.1.
Set Y := (LiT,Y2T), Qy := 02 (Y), and Vy:= (ZY). Observe (Y,LiT,Y 2 T) is
a Goldschmidt triple, so (LiT/Qy,T/Qy,K 2 T/Qy) is a Goldschmidt amalgam by
F.6.5.1, and hence is listed in F.6.5.2. Now K 2 = L2 has at least three noncentral
2-chief factors in L; so as this does not hold in any case in F.6.5.2, we conclude
Qy-!=-1, so that YE 1-l(T). Hence Vy E R 2 (Y) by B.2.14.
We saw Y 2 i. M, so Yi. M. On the other hand, for z E Vi - Zv, Cy(Vy) :::;
Cy(Zv) n Cy(z):::; CM(z), so applying 13.3.9 with Y, 02 (Cy(Vy)) in the roles of
"H, Y", and recalling that L/0 2 (L) ~ A 6 , we conclude that 02 (Cy(Vy)) = 1 or
Li.
In the latter case, Y2 acts on Li, and hence centralizes Lif 02 (Li) so that
Li normalizes 02 (Y202(Li)) = Y 2. Then as L2 = K2, L = (Li, L 2 ) :::; Na(Y2),
contrary to Y2 i. M = !M(LT). Thus 02 (Cy(Vy)) = 1 so that Cy(Vy) = Qy:::;
03 1(Y). In addition this argument shows that [Li, Y2] i. 02(Li).
Let Y := Y/Cy(Vy) and y+ := Y/0 31 (Y), so that y+ is a quotient ofY, and
is described in F.6.11.2. Now L = [L, J(T)] by 13.4.3.1, so that Li = [Li, J(T)]
by 13.2.2.4; so as J(T) centralizes 03 (F(Y)) by Theorem B.5.6, so does Li.
In particular Li centralizes F ( 031 (Y)), so as Li is generated by conjugates of
an element of order 3, we conclude from A.1.9 that Li :::; Cy(03'(Y)). Thus
Li= 031 (Li031(Y)), so if [Y 2 +,LtJ :::; 02(Lt), then [Y2,Li] :::; 02(Li), which
we showed earlier is not the case. We conclude [Y 2 +,LtJ i. 02 (Lt). Now as
J(T) :::) Y2T since we are in case (iii), but Li= [Li, J(T)], 02(Y2T)-!=-02(LiT).
Thus case (i) of F.6.11.2 holds, so y+ is described in F.6.18, where F.6.18.1 is
similarly ruled out. As Li = [Li, J(T)], y+ is not E 4 /31+^2 by Theorem B.5.6,
while the condition [Y/, Lt] i. 02(Lt) rules out the other possibility in F.6.18.2.
In the remaining cases in Theorem F.6.18, Y is not solvable, so there is Ky E C(Y),
and by 13.4.5, Ky/02(Ky) ~ A5, L3(2), A 6 , or A5. The A5 case is ruled out, as
A5 does not appear as a composition factor in the groups listed in Theorem F.6.18.
Similarly conclusion (3) of F.6.18 does not hold, so Li :::; Ky.
As Li = [Li, J(T)], K;, = [Ky, J(T)], so by Theorem B.5.1 and 13.4.5.3,
[Vy, Ky] is a natural module for KY. ~ L 3 (2) or A5, a 5-dimensional module for
KY. ~ A 6 , or the sum of two natural modules for KY. ~ L 3 (2). As Z =Vi is of
rank 2, with (zY^2 ) ~ E 16 , Vy is the sum of two natural modules for KY. ~ L 3 (2).
As LiT* is the parabolic of KY. centralizing Z, J(Ri):::; Cy(Vy) = Qy, and hence
Baum(Ri) = Baum(Qy) by B.2.3.5. Then each nontrivial characteristic subgroup