1.2. THE SET .C(G, T) OF NONSOLVABLE UNIQUENESS SUBGROUPS 507Assume as in (2) that L -=J L8, and that L < K; then Ks -=J K by (1). Then
by 1.2.1.3, K/02(K) is L2(2n), Sz(2n), L2(p), or Ji, and L/0 2 (L) is also in this
list. Consulting A.3.12, we see the only possible proper embeddings of Lin Kare
those given in (2). This establishes (2) and (3).
Finally ( 4) is established similarly: from the list of groups in Theorem C, weextract the sublist not occurring as an initial possibility in A.3.12. D
We next wish to study the action of members of .C( G, T) on their internal
modules. To do so, we use some of the results from section A.4 of Volume I. Recallfrom Definition A.4.5 that X consists of the nontrivial subgroups Y of G satisfying
Y = 02 (Y) and F*(Y) = 02 (Y). Notice the second condition says that X s;;; 'He.
Now for L E .C( G, T), L = L°^0 by the definition of C-component, while L E 'He
by 1.2.7.2, so that .C(G, T) S: X. Next recall that for YE X and RE l!INa(Y) (Y, 2),
from Definition A.4.6V(Y, R) := [01(Z(R)), Y] and V(Y) := V(Y, 02 (Y)).
There we also defined Xt to consist of those YE X with V(Y) # 1. The subscript
"!" stands for "faithful"; for example, if X E Xt with X/02(X) simple, then
X/0 2 (X) is faithful on the module V(X). Define.C1(G, T) := .C(G, T) n Xf,
and also define
.Cj(G,T) :=L*(G,T)nXf,
which of course coincides with .C1(G, T) n C*(G, T). Now by definition, elementsof .Cj(G, T) are maximal in the subposet .C1(G, T); in the next lemma we see that
the converse holds.LEMMA 1.2.9. Let LE .C1(G,T). Then
{1) If L::::; KE .C(G, T), then V(L, 02(Nr(L)L)) ::::; V(K, 02(Nr(K)K)), and
so KE .Ct(G, T).{ 2) If L is maximal in £, f ( G, T) with respect to inclusion, then L E £,* ( G, T),
and hence LE .Cj(G, T).
PROOF. Let L ::::; K E .C(G, T) and R := Nr(L). By 1.2.8.1, R = Nr(K).
Thus RE Sylz(NKR(L)), so 02(KR)::::; R by A.1.6, and 02(RL) = CR(L/02(L)).
Hence we may apply parts (2) and (3) of A.4.10 to obtain (1). Then (1) implies
(~. DLEMMA 1.2.10. Let TE Sylz(G), HE 'H(T), and LE C(H). Then the follow-
ing are equivalent:
(1) LE .C1(G,T).
(2) There is VE R2(H) with [V, L] # 1.
{3) [R2(H), L] # 1.
In particular the result applies to LE .C*(G,T) and HE 'H((L,T)).PROOF. We have F*(H) = 02(H) by 1.1.4.6, and from 1.2 .. 1.4 we see that all
·non-central 2-chief factors of L lie in 02(L). These are the hypotheses for A.4.11,
whose conclusions are exactly the assertions of 1.2.10. D