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

140

PRIME-POWER GROUPS

[CH. V

this regular group, and the group of isomorphisms of a regular
group is the subgroup composed of all the substitutions which
omit a given letter in its holomorph. Hence we may reduce
each of the substitutions which transform H in a certain way
to one of the given form by transforming by a substitution in
the direct product of the conjoints of H1, H2,..., Hp.
From the preceding paragraph it results that the number
of groups of order pm which contain a given group H of order
pm-1 cannot exceed the order of the product of H and the
order of its group of isomorphisms. Hence the number of the
groups of order p" is always finite when p is finite. We shall
be able, however, to reduce this apparently possible number
very greatly. In the first place it should be observed that if
sl' is so selected that 1'2S3... Sp is commutative with t,
while sl' involves only letters in H1, then sl=sl"sl' where
Si" is commutative with Si'; otherwise (sIS2... s.sp) would
not be in H. This restricts the number of choices of si" to the
number of operators in H which are both in its conjoint and
also invariant under si. Another important restriction is that
we evidently need to consider only one of the first p-1 powers
of Sl1S2S3.. Sp, since t may be transformed into any power
without affecting sls2... Sp by permuting the constituents
H2, H3,.., H Hp cyclically. When H is abelian, all the groups
obtained by replacing si" by any of its powers prime to its order
are conjugate, since the operators in the group of isomorphisms
of an abelian group which transform each operator of this group
into the same power are invariant under this group of isomorphisms.
To illustrate the method outlined above we proceed to
determine the possible non-abelian groups of order p3, p>2.
It has been observed that each of these groups involves a noncyclic group of order p2, and hence we may use this for H. As
the group of isomorphisms of H involves only one set of conjugate subgroups of order p, the two substitutions s 'S2S3... sp
and t may be supposed to be the same for all these groups.
Moreover, Si" is restricted to a single subgroup, and hence it
is necessary to consider only two cases, viz., the case when