574 3. DETERMINING THE CASES FOR LE C'j(G, T)Hypothesis 3.1.5, V E R 2 (M 0 ), so V is elementary abelian, normal in T, and
contained in D1(Z(0 2 (M 0 ))). Further T:::; M 0 :::; M so that 02(M):::; 02(Mo)
by A.1.6; and M E M(T) ~ He since G is of even characteristic. Therefore
V:::; CM(02(M)) :::; 02(M), and hence V :::; 02(H n M.). Finally V i. 02(H) in
case (i), and 02 (H) = kerHnM(H) by B.6.8.5. This completes the verification of thehypothesis of E.2.13. Hence we conclude from E.2.13.3, that q(AutH(V), V) :::; 2.
Therefore since T is Sylow in both H and Mo, q(Mo/CM 0 (V), V) :::; 2. Further
if V is a TI-set under M, then we have the hypotheses for E.2.15, so that result
shows that q(AutH(V), V) < 2, and hence q(M 0 /CM 0 (V), V) < 2. Thus (2) holds,
as claimed.Thus we may assume that cases (i)-(iii) do not hold. In particular, case (iv)
holds; and as (i) fails, now V :::; 02 (H). By our observation following Hypothesis
3.1.5, it suffices to prove that RE Syb(0^2 (H)R), since then Theorem 3.1.1 showsthat conclusion (1) of Theorem 3.1.6 holds.
Set QH := 02(H), K := 02 (H), and H* := H/QH. As case (iv) holds,
Q := Rn QH ::::l H, so as Cr(V) = Rand V :::; QH by the previous paragraph,
V:::; Z(Q). Therefore U:::; Z(Q). ·
We saw earlier that J(T) = J(R) i. QH, and H is a minimal parabolic
described in B.6.8. Now by Hypothesis 3.1.5, QH :::; T :::; Mo :::; Na(R), so·[QH, J(R)] :::; QH n R = Q, and hence [K, J(R)]J(R) centralizes QH /Q. Next
[K, J(R)]J(R) is normal in KT= H, but J(R) i. QH, so K:::; [K, J(R)]J(R) by
B.6.8.4, and then K centralizes QH/Q. Therefore [02(K), K]:::; Q.
If K centralizes U then K centralizes V, so Cr(V) = R is Sylow in Ca(V) and
hence R is Sylow in KR, which as we observed earlier suffices to complete the proof.
Thus we may assume that K does not centralize U. Then CH(U) :::; kerHnM(H)
and Cr(U) = CqH (U) by B.6.8.6.
As J(R) i. QH, there is some A E A(R) with A* -=f. 1. As A:::; Rand U:::; Z(Q),AnQH = AnQ:::; CA(U), so AnQH = CA(U) by the previous paragraph. Then as
A E A(R), r A*,U :::; 1 by B.2.4.1. Now U might not be in R 2 (H), but each nontrivial
H-chief section Won U is an irreducible for H/CH(W), so that 02 (H/CH(W)) =
1. Furthermore CH(W) :::; kerHnM(H) and Cr(W) = CqH(W) by B.6.8.6, so
m(A*) = m(AutA(W)) and hence rAutA(W),W:::; TA•,u:::; 1. Therefore Wis an
FF-module for AutH(W). Hence by B.6.9 and E.2.3, m(W/Cw(A)) = m(A),
K = Ki or KiK2, and [W, Ki] is the natural module for Kl ~ L 2 (2n), A 3 , or
A5· Furthermore as m(U/Cu(A)):::; m(A*) = m(W/Cw(A*)), we conclude Ki has
a unique noncentral chief factor Ui on U, where Ui = Ui/Cui (Ki) is the natural
module for Kl, and [U, Ki] = Ui·Set B := H n M and observe that B is solvable: This is clear if H is solvable,
while if H is not solvable then by E.2.2 and the previous paragraph, B n K is a
Borel subgroup of K*, and in particular B is solvable. By Hypothesis 3.1.5, either
(I) holds and B normalizes L := 02 (M 0 ), or (II) holds and B normalizes V. In case
(I), let D := CB(L/02(L)), and in case (II), let D := CB(V). Then B normalizes
D in either case.We claim that R is Sylow in D, and D ::::l B: In case (II), R = Cr(V) is Sylow
in Ca(V), and hence also in CB(V) = D. As B normalizes V in (II), D ::::l B.
In case (I), we apply parts (4) and (5) of A.4.2 with L, Mo in the roles of "X,
M", to see that R = 02(Mo) is Sylow in CMo (L/02(L)). Hence R is also Sylow in
CB(L/02(L)) = D. As B normalizes Lin (I), D ::::l B.