2 PART III, CHAPTER 1. THEOREM C7: GENERAL INTRODUCTION
(c) One of the following holds:
(1) O
p
/ ( M ) ^ l ; o r
(2) M = NG(K) for some K G such that K G Qhev{2) and m
p
(C
G
(i^) )
1.
Note that it is immediate from [I2; 8.7] that subgroups M satisfying this defi-
nition are in fact strong p-uniqueness subgroups.
In this part we shall thus replace the condition p G cr0(G) by the weaker con-
dition:
, , If p is odd, then G possesses no strong p-uniqueness subgroup of compo-
nent type.
Recall that by definition, M is a p-component preuniqueness subgroup of G if
and only if M is proper subgroup of G and M possesses a p-component K such
that CG(X) M for every element x G CM{K/OP(K)) of order p (it is allowed
that CM{K/OP'(K)) be a p'-group). In this situation we say that M is a K-
preuniqueness subgroup. The term "almost strongly p-embedded" in turn means
that one of four conditions is satisfied [I2; 8.1-8.4]: M is strongly p-embedded in
G, M is wreathed, M is almost p-constrained, or M is of strongly closed type. The
first three conditions emerge from application of our p-uniqueness subgroups [II3;
pp. 134-135]. The fourth condition will emerge separately from our analysis. We
add to it here one further configuration beyond that defined in [I2; Definition 8.3],
since the two configurations are better handled together as part of the uniqueness
case. Thus we relax that definition1 as follows.
D E F I N I T I O N 1.2. Let M be a if-preuniqueness subgroup of G and let P G
Sylp(M) and Q = CP(K/Op(K)). Then M is of strongly closed type if and only if
(1) p is odd;
(2) Either K G Qhev(2) with mp{Q) = 1, or K/Op
p
(K) ^ Lep(q), e = ± 1 , q = e
mod p with Q,i(Q) K;
(3) Qi(Q) (if nontrivial) is strongly closed in P with respect to G;
(4) NG(X) M for any noncyclic X P and for any 1 ^ X P for which
mp(CP(X)) 3.
The technical notion of strong p-uniqueness subgroup is only relevant when p
is odd; when p = 2, we freely use the more decisive uniqueness theorems in [II2].
For each prime p the set Sp (or the "set of Sp-groups") is defined in [I2; 13.1] as
the complement of a certain subset of the set %p of all quasisimple X-groups such
that Op'(K) 1. Specifically, a simple X-group K of order divisible by p is called
a Sp-group if and only if
(1) K £ Qhev(p) and K ^ Api A2p or A3p;
(2) mp(K) 2;
(3) If p = 2, K is not isomorphic to L,2(q), q odd, q 5, 1/3(3), 14(3),
e = ± 1 , G
2
(3), A7, M n , M12, M
2 2
, M
2 3
, M
2 4
, J
2
, J3, ^4, i ? 5 ,
(1C) Suz, Ru, Colj Co2, Fi22, Fi23, Fi24, F
3
, F
2
,or F
x
;
(4) If p = 3, K is not isomorphic to £!((?), Q ' = e mod 3, e =
± 1 , C/5(2), f/6(2), 5p
6
(2), £4(2), 3Z34(2), F
4
(2), 2 F
4
(25)', Sp
4
(8),
G
2
(8), A
7
, M n , M12, M
2 2
, J
2
, J3, C01, Co
2
, Co
3
, 5w2, Mc , Ly,
ON, Fi22, Fi23, Fi
2 4
, F5, F3, F2, or F
i ;
1
It is the possibility of the Lp(q) configuration in (2) which has been added here.
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