2. SOME DC-GROUP CONDITIONS

5

one of these properties in its proof, either the statement of the theorem will ex-

plicitly indicate that fact, or the whole section in which the theorem occurs will be

under a blanket assumption, as in Sections 20, 21, and 22.

For reference we gather all such properties into this section.

The first few properties are relevant to Lp -balance and the study of signalizers.

DEFINITION

2.1. The simple group K has the Schreier property if and only

if Out (if) is solvable. If X is any group, then X has property (S) if and only if

every simple section of X has the Schreier property.

DEFINITION

2.2. Let p be a prime. The simple group K has the p-Schreier

property if and only if for any P G Sylp(K), Cxui(K){P) is p-solvable. If X is any

group, then X has property (Sp) if and only if every simple section of X has the

p-Schreier property.

DEFINITION

2.3. If p is a prime, the simple group K has the p-Glauber man

property if and only if CAut(K)(P) has a normal p-complement for any P G

Sylp(K). If X is any group, then X has property (Gp) if and only if every

simple section of X has the p-Glauberman property.

REMARK.

By Sylow's theorem, these conditions are independent of the choice

of P. Further, it is immediate that the p-Glauberman property implies the p-

Schreier property, so (Gp) implies (Sp). As remarked above, Glauberman has proved

(G2) for all groups. The proof is based on his Z*-Theorem and underlay the original

proof by Gorenstein and Walter of the L-balance theorem [GW4].

The next definition makes use of the notions of p-component and quasisimplicity

(see Section 3).

DEFINITION

2.4. Let p be a prime. The simple group K is said to have the Bp-

property if and only if for any automorphism x of K of order p, every p-component

of CK(X) is quasisimple. A group X is said to have property (Bp) if and only if

every simple section of X has the Bp-property.

For p = 2 & proof of the ^-property for all simple groups has been given by

Walter [Wa3]. This is independent of the the full classification theorem but has a

classification-theoretic flavor.

The next properties are needed for McBride's proof of the nonsolvable signalizer

functor theorem [McBl,McB2].

DEFINITION

2.5. Let p be a prime. The simple group K is outer p-cyclic if

and only if K is a p'-group and A\xt(K) has cyclic Sylow p-subgroups. The group

X has property (Cp) if and only if every simple section of X which is a p'-group

is outer p-cyclic.

DEFINITION

2.6. Let p be a prime. The simple group K has the p-McBride

property if and only if

(i) p does not divide \K\\

(ii) K is outer p-cyclic; and

(iii) Either any noncyclic simple section of K is isomorphic to one of the groups

L2(2p), Sz(2p), U3(2p),

or

L2(3P),

or else for any automorphism a; of If of

order p, the centralizer C = CK{X) has the following properties:

(a) C normalizes no nontrivial solvable subgroup of K;