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.