15.3. THE ELIMINATION OF Mr/CMr(V(Mr)) = 83 wr Z 2 1125(2) Assume that m3(L) = 1 and Y+ induces inner automorphisms of L/0 2 (L).
Then Y+ = YLYe where Ye := 02 (CY+ (L/02(L))), IYLl3 = 3 = IYel, andY+/02(Y+) ~ Eg.
PROOF. First Y+ = 02 (Y+) normalizes L by 1.2.1.3. Then m 3 (LY+) ::::; 2 since
I is an SQTK-group, so as m3(Y+) = 2 and m 3 (L) ~ 1 by hypothesis, YL-=/:-1. As[YL, R] is a 2-group, R normalizes L by 1.2.1.3. Further 02 (L)0 2 (YL) is normalized
by Y+, and so lies in R, completing the proof of (1).
Assume that m3(L) = 1. As YL-=/:- 1 by (1) while Y+ is of exponent 3, IYLl 3 = 3.
Assume also that Y+ induces inner automorphisms on L/0 2 (L); then as m 3 (L) = 1
and Y+ is of exponent 3, Auty+(L) = AutYL(L). Hence Y+ = YLYe, with YL n Ye
a 2-group as Z(L/0 2 (L)) = 1. In particular, Y+/0 2 (Y+) ~ E 9 rather than 31 +2,
and 1Yel3 = 3, completing the proof of (2). D
We now begin our analysis of the case F*(I) = 02 (1). Observe then that
Ur = (Zr) E R2(J) by B.2.14.
We partition the problem into the subcases m 3 (Y+) = 2 and m 3 (Y+) = 1.THEOREM 15.3.13. Assume IE 1-l+ and m 3 (Y+) = 2. Then F*(I)-=/:- 02 (1).
Until the proof of Theorem 15.3.13 is complete, assume I is a counterexample.Then F*(I) = 02 (1), so that Ur E R2(I) by B.2.14. As m3(Y+) = 2, case (i) of
15.3.11.5 holds, so Y/0 2 (Y) ~ _E 9 or 3i+^2 , and R = Gr(V). If case (2) of Hypothe-
sis 15.3.10 holds, then as Y+ < Y, Y/0 2 (Y) ~ 31+^2 and Gy(V)/0 2 (Gy(V)) ~ Z 3.
Thus:LEMMA 15.3.14. {1) V :S Z(R).
(2) If case (2) of Hypothesis 15.3.10 holds, then Y/02(Y) ~ 3i+^2 and !Gy(V):
02(Cy(V))i = 3.LEMMA 15.3.15. {1) Hypothesis C.2.3 is satisfied with I, Mr in the roles of
''H, MH"·
{2) There exists L E c;,(J(J)) with L i. Mr, m3(L) ~ 1, L = [L, Y+J, and L*
and L / 02 ( L) quasisimple.
{3) Each solvable Y+S-invariant subgroup of I is contained in Mr.
PROOF. As case (i) of 15.3.11.5 holds with R = Gr(V), (1) follows. By
15.3.11.11, J(I) i. Mj. In particular J(I) -=/:-1, so that Ur is an FF-module
for I by B.2.7, and hence J(I) is described in Theorem B.5.6. If L is a di-
rect factor of J(I)* isomorphic to 83 , then there are at most two such factors by
Theorem B.5.6, so y_t' = 02 (Y-t') normalizes and hence centralizes L, and then
L ::::; Nr• (Y-t') = Mj by 15.3.11.9. Thus as J(I)* i. Mj, Theorem B.5.6 says there
exists L E C(J(J)) with L i. Mr, m 3 (L) ~ 1, and L* quasisimple. By 1.2.1.3,
Y+ = 02 (Y+)::::; Nr(L). By 15.3.11.2, Cr(Ur)::::; Mr, so as Li. Mr, L = [L, Y+] by
15.3.11.9. Further as L* is quasisimple and not isomorphic to Sz(2m) by Theorem.
B.5.6, 03 ,(L):::; Gr(Ur):::; Mr, so by 15.3.11.6, [03'(L), Y+]:::; Y+n031(L):::; 02(L),
and hence L/0 2 (L) is quasisimple by 1.2.1.4. Thus (2) holds. Then by 1.2.1.1, each
member of 1-l+ is nonsolvable, so (3) follows. DDuring the remainder of the proof of Theorem 15.3.13, pick Las in 15.3.15.2.
Set YL := 02 (Y+ nL), Ye:= 02 (0y+(L/02(L)), SL:= SnL, RL := RnL, and
ML :=Mn L. Let WL := (VL) and (LRY+)+ := LRY+/GLRY+ (WL)·