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

~50] NUMBER OF THEIR SUBGROUPS 123
5. Every non-abelian group of order pm contains an invariant commutator of order p.
6. If a group of order 3m contains no operator of order 9 all of its operators in any complete set of conjugates are commutative.*
Suggestion: If Si, s2 are any two operators of such a group it results
that (SS2)3 = S1 ' S2S1S22 'S 2S1S2 = (SlS22)3 =-S1' S22' S2' S122= 1.
7. A necessary and sufficient condition that a group of order pm is abelian is that more than pm-l of its operators corresponds to their inverses
in some automorphism of the group.
8. If two non-commutative operators of a group of order pmr, p>2,
correspond to their inverses in an automorphism of the group their commutator cannot correspond to its inverse in this automorphism.
50. Number of Subgroups in a Group of Order pt. We
shall first determine the form of the number of subgroups of
order pm"- in a group G of order pm. Any two subgroups
of order pm-l must have pm-2 operators in common, and these
common operators constitute a group which is invariant under G.
They must therefore include all the commutators of G and also
the pth powers of every operator of G. If H is composed of all
the operators which are common to all the subgroups of order
p"-l contained in G it must include all the commutators of
G as well as the pth powers of all its operators. From this
it results that the quotient group corresponding to H is abelian
and of type (1, 1, 1,... ). Each subgroup of order pm-l
in G must correspond to a subgroup of index p in this quotient
group. In Chapter IV, ~ 40, we proved that the number of these
subgroups of index p is equal to the number of the subgroups
of order p in this quotient group. Hence the theorem:
The number of subgroups of order pm-l contained in a group
G of order pm is always of the form
p-1
P -
To obtain the exact number of these subgroups it is necessary to observe that pX is the order of the quotient group
of G with respect to the group formed by all of its operators

* W. Burnside, Quarterly Journal of Mathematics, vol. 33 (1901), p. 231.