506 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS
We will focus primarily on the case where the role of Sis played by T E Syb( G).
In this case when HE H(T), then T. is also Sylow in H, so an earlier remark now
specializes to:
LEMMA 1.2.6. C(H) ~ .C(H,T) ~ .C(G,T) for each HE H(T).
THEOREM 1.2.7 (Nonsolvable Uniqueness Groups). If LE £*(G, T) then
{1) LE C(H) for each HE H((L,T)).
{2) F*(L) = 02(L).{3) Ne( (LT)) = !M( (L, T)).
(4) Set Lo := (LT) and Z := 01(Z(T)) Then Cz(Lo) n Cz(Lo)^9 = 1 for
g E G - Nc((LT)).
PROOF. Let HE H((L,T)). As TE Syb(G), TE Syb(H), so also LE
.C(H, T). Then by 1.2.4, L ~KE C(H) for some K. But by 1.2.6, C(H) ~ .C(G, T);
so L = K from the maximal choice of L. Hence (1) holds.
Next by 1.1.4.6, F*(H) = 02 (H); so as L is subnormal in H, (2) holds by
1.1.3.1.
Set Lo:= (LT). As LE C(H), Lo :::] Ji by 1.2.1.3. Hence H ~ M := Nc(L 0 ),
and as 02 (L) I= 1 by (2), 02 (M) I= 1. In particular if H E M(T), we conclude
H = .M. Thus (3) holds. ,To prove (4), assume Zo := Cz(Lo)nCz(Lo)^9 I= 1. Then LoT, L5T^9 ~ Cc(Zo),
so using (3), M = !M(Cc(Z 0 )) = MB; but then g E Na(M) = M as M E M,
contrary to g tj. M. D
Part (3) of 1.2.7 says that if LE .C*(G, T) then (L, T) is a uniqueness subgroup
of G. This fact plays a crucial role through most of our work.
Next we obtain some further restrictions on chains in the poset .C(G, T). For
example we see in part (4) of 1.2.8 that for many choices of L/0 2 (L), LE .C(G, T)
is already maximal. In parts (2) and (3) of 1.2.8 we see that if Lis not T-invariant,
then usually L is maximal.
LEMMA 1.2.8. Let S be a 2-subgroup of G, and L,K E .C(G,S) with L ~ K.
Then
{1) Ns(L) = Ns(K). So if LI= L^8 then LL^8 ~ KK^8 for KI= K^8 •{2) If L < (Ls), then either
(a) L = K, or
{b) L/02(L) ~ A5, and K/02(K) is either J1 or L 2 (p) for some prime p
with p^2 = 1 mod 5.
{3) If L <(Ls), then either LE £(G, S), or L/0 2 (L) ~ A 5.
(4) We have L E .C(G, S) if L/02(L) is any of the following: A 7 ; L 2 (r^2 ),
r > 3 an odd prime; (S)L3(p), p an odd prime; Mu, Mi2, M23, J1, J2, J4, HS,
He, Ru, L5(2), or (S)U3(2n); a group of Lie type of characteristic 2 and Lie rank
2, other than L3(2) or L3(4).PROOF. Let H := (K, S), and recall C(H) = {K} or {K, K^8 }. By I°.2.4, K
is the unique C-component of H containing L, so that Ns(L) ~ Ns(K). The
opposite inclusion follows from A.3.12, as we check that in each of the embeddings
listed there, K does not contain a product of two copies of L, so that Lis Ns(K)-
invariant. Hence (1) holds.