1.2. THE SET .C*(G, T) OF NONSOLVABLE UNIQUENESS SUBGROUPS 505Next we extend the notation of .C(X, Y) in Definition A.3.10 to our QTKE-
group G. This will help us keep track of the possible embeddings of C-components
of a subgroup Hi E 1i in some other Hz EH, as long as H 1 and H 2 share a common
Sylow 2-subgroup.DEFINITION 1.2.3. For H a finite group, and S a 2-subgroup of H, let .C(H, S)be the set of subgroups L of H such that
(1) LE C((L,S)),
(2) S E Syl2( (L, S) ), and(3) 02((L,S)) -j.1; that is, (L,S) E 1iH.
Assume for the moment that H E H, S E Sylz(H), fI := H/0 2 (H), and
L E .C(H, S). Then by Hypotheses (QT) and (K), fI satisfies Hypothesis A.3.4,
with S E Syl2(H); so from condition (1) of the definition of .C(H, S) and A.3.3,
L E .C(fI, S), defined only for fI in section A.3. Also applying 1.2.1.3 to (L, S),
either Ls = L and (L, S) = LS, or Ls = {L, L^8 } and (L, S) = LL8 S. Further as
in A.3.11, C(H) ~ .C(H, S), so when C(H) is nonempty, .C(H, S) is nonempty.
Now just as in section A.3, we wish to see how members of .C(H, S) embed in
H.
LEMMA 1.2.4. Let HEH, with SE Sylz(H); set fI := H/0 2 (H), and assume
BE .C(H,S). Then B:::; L for a unique LE C(H), and the pair (B,L) is on the
list of lemma A.3.12. In particular
(+)If S normalizes B, then L ::;J H.
PROOF. We apply A.3.12 to conclude B is contained in a unique L E C(fI),with the pair (B, L) on the list of A.3.12. Then using the one-to-one correspondence
from A.3.3.4, L is the image of a unique L E C(H); and as B :::; 02(H)L we see
B = B^00 :::; (0 2 (H)L)^00 = L. This completes the proof, as (+)follows from the
uniqueness of L. D
LEMMA 1.2.5. Let HEH, SE Sylz(H), RS S with IS: RI= 2, and suppose
LE .C(H, R). Then there exists a unique KE C(H) with L:::; K.
PROOF. The proof is much like that of A.3.12. Let H := H/0 00 (H). By
1.2.1.1, H^00 =Ki··· Kr where Ki E C(H), and by 1.2.1.2, H^00 =Ki x · · · x K;.
Now L = L^00 :::; H^00 , so for some i (which we now fix), the projection P of L on
K := K:t is nontrivial. As P is a homomorphic image of L E C(L), P E C(P)
by A.3.3.4.
As SE Sylz(H) and K is subnormal in H, Sn K E Sylz(K), and similarly
Rn L E Syl 2 (L) using our hypothesis that L E .C(H, R). Then as R ::; S, Sn
L = Rn L E Syl 2 (L), so Sn P E Sylz(P), for P the preimage of P. Then
ISnP: Rn Pl SIS: RI S 2; so (RnP) i Ooo(P), as otherwise P /Ooo(P) has
Sylow 2-groups of order at most 2, and so is solvable using Cyclic Sylow 2-Subgroups
A.1.38, contrary to P E C(P) nonsolvable. Hence [L, Rn P] i 000 (L). However
as (Rn P) acts on P* and permutes the C-components LR of (L, R), Rn Pacts
on L; so by A.3.3.7, L = [L, Rn P] :::; [L, K] :::; K. Finally K is unique since
Kin Kj :::; 000 (H) for any j "I-i. This completes the proof of 1.2.5. D
Lemma 1.2.4 gives information about .C(H, S) considered as a set partially
ordered by inclusion. This leads us to define .C*(H, S) to be the maximal members
of this poset.