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

~ 551 CONSTRUCTION OF GROUPS OF ORDER pm

141

si" is the identity and the case when its order is p. That is,
there cannot be more than two non-abelian groups of order
p3, p> 2, in accord with the general theory which precedes
this paragraph. From the present paragraph it results that
one of these groups involves operators of order p2 while the
other does not have this property. Hence there are two and
only two non-abelian groups of order p3, p> 2. It was proved
in ~ 33 that there are also only two such groups when p =2.
Among the groups of order pm which involve H the one generated by H and t is of especial interest in view of its simple
structure. In fact, it is abstractly the direct product of H and
the group of order p. If we transform this group by
rlP-lr3r42.. rp-2
where rl is any invariant substitution of H1 while ra, = 3,...
p, is the transform of rl with respect to t", there results a group
generated-by H and ri-'Pt. From this and the preceding theory
it results that we need to use only two values for si" when
H is cyclic and the entire group is abelian, as results also directly
from the theory of the abelian groups. This general method
can easily be extended so as to reduce the amount of labor
necessary to determine all the possible groups of order pm
which involve a given subgroup of order pm-l *; but the preceding developments may suffice to point the way towards a
penetration into this difficult subject. From the given illustration it results that we do not always need to use all the possible groups of order pm-l for H in order to determine all the
groups of order pm. In fact, when m=4 we need to consider
only the abelian groups of order p3 for H, since every group of
order p4 contains an abelian subgroup of order p3.
EXERCISES
1. Determine the fourteen possible groups of order 16.
2. Determine the non-abelian groups of order p4, p>2, which involve
no operator of order p2. Show that si"= 1 in these cases and that there
are two such groups when p>3, one having a commutator subgroup of
* American Journal of Mathematics, vol. 24 (1902), p. 395.