Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 491

INVARIANT ABELIAN SUBGROUPS

121

whose central is of order at least p -2, whenever m- (3-1)
(/~-2)/2. When m = 3( -1)/2 the order of this subgroup is
p#(#- 1)/2 - (3- 2) (t- 3)/2 = p2 - 3.
The quotient group with respect to the given central is of order
p-~1. If this quotient group contains operators of order p2,
G must evidently involve an abelian group of order pa. It
remains therefore to consider the case when this quotient
group does not involve any operator of order p2. If p = 2 we may
assume that this quotient group is abelian, and hence we shall
confine our attention, in what follows, to this special case.
We are thus led to consider the possibility of constructing
a group K of order 22f-3, having a central C of order 2a-2 which
leads to an abelian quotient group of type (1, 1, 1,.. ).
If we arrive at a contradiction by assuming that K does not
include an abelian subgroup of order 20 our theorem is proved.
IfK existed, all the operators of C besides the identity would
be of order 2, since all of these operators would be commutators
of K. Moreover, each of the non-invariant operators of K
would be transformed under K into itself multiplied by all
the operators of C.
Let K1 represent any subgroup of order 220-4 and involving
C. Each of the non-invariant operators of K1 is transformed
under K1 into itself multiplied by all the operators of a subgroup
of order 20-3 contained in C. The multiplying subgroups for
two distinct operators (mod C) of K1 must be distinct, otherwise the operators of the group of order 4 (mod C) generated
by these two operators would have to be transformed, by an
operator of K which is not also in K1, into themselves multiplied by the operators of a group of order 4 which has only the
identity in common with the given subgroup of order 2~-3
in C. As this is clearly impossible it results that all the different non-invariant operators of K1 are transformed under
K1 into themselves multiplied by all the different subgroups of
order 2a-3 in C.
From the preceding paragraph it results that there is a (1, 1)
correspondence between the operators of K1 and the subgroups