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;
Previous Page Next Page