1.4. PROPERTIES OF SOME UNIQUENESS SUBGROUPS{2) Q E l!IG(Lo, 2) = Syb(CM(Lo/02(Lo))).
{3) Ca(Q) ~ 02(M) ~ Q.
(4) If LE Xf, then there is VE R2(LoT) with [V, Lo] -=f. 1.
515(5) If LE .C:j(G, T), assume that L/02(L) is quasisimple. Let VE R2(Lo, T)
with [V, Lo] -=f. 1. Then CL 0 r(V) ~ 02,w(LoT), Cr(V) = Q, and fh(Z(Q)) =R2(LoT).
PROOF. First M := Na(Lo) = !M(LoT) by 1.2.7.3 or 1.3.7. Then since LoT ~
Na(Q) by definition of Q, also M = !M(Na(Q)).
Next if 1 -=f. R char Q, then embedding Na(R) ~NE M, we have Na(Q) ~
Na(R) ~ N, forcing N = M as M = !M(LoT). So (1) holds.
Now (2) follows from A.4.2.7. By (1), Ca(Q) ~ M, and by (2), 02(M) ~ Q.
Also M E H(T) ~ He by 1.1.4.6, so
Ca(Q) ~ CM(Q) ~ OM(02(M)) ~ 02(M),
giving (3).
Next when LE .C1(G, T), there exists VE R 2 (L 0 T) with [V, L] -=f. 1by1.2.10.3,
while this follows from A.4.11 when L E 31( G, T), since there all 2-chief factors lie
in 02 (L). Thus (4) holds. Finally assume that either LE Sj(G, T) or LE .Cj(G, T)
with L/0 2 (L) quasisimple. Therefore LoQ/02,w(LoT) = F*(LoT/02,w(LoT)) is a
chief factor for LoT, so as [V, Lo] -=f. 1, CL 0 r(V) ~ 02,w(LoT). But Q is Sylow in
02 ,w(L 0 T) so Cr(V) ~ Q, while as Vis 2-reduced, Q = 02(LoT) ~ Cr(V). This
completes the proof of (5). D