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

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NUMBER OF THEIR SUBGROUPS

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group of order p"+l as many times as it contains different
subgroups of order p" and thus arrive at the following equation:
x=ra = r,a+l
Z sx= 2 sy.
x=l y=l
As both Sx and sy are congruent to unity modulo p it results
that r-ra-+i(mod p). As ri=l(mod p) it results that r,-1
(mod p) and hence the number of the subgroups of order p" in a
group of order pm is always of the form 1 +kp.
It is now very easy to prove that the number of the subgroups of order pa in any group G' whose order is divisible by
pa is always of the form 1+kp, even if the order of G' is not a
power of p. If a subgroup of order pa in G' is not invariant
under some one of the Sylow subgroups of order pm in G' it
evidently belongs to a complete set of ps conjugates. Hence
we may confine ourselves to these subgroups of order p" in G'
which are invariant under a particular subgroup of order pm
in G'. All of these must occur in this Sylow subgroup of order
pm and hence their number is of the form 1 +kp. This proves
that the total number of the subgroups of order pa must also
be of this form, so that we have proved the following theorem
due to Frobenius:
The total number of the subgroups of order pa in any group
whose order is divisible by pa is of the form 1 +kp.
This theorem holds whether the order of the group is or is
not a power of a single prime, and it may be regarded as an
extension of Sylow's theorem. It should, however, be observed
that the subgroups of order p" are not always conjugate when
they are not Sylow subgroups.
If the group G of order pm contains at least one abelian
subgroup of order p3 it is easy to prove that the number of its
abelian subgroups of order p3 is of the form 1+kp. We proceed
to prove this theorem. If G contains an abelian subgroup
H of order p" the totality of the operators of G which are commutative with every operator of H forms a group H' which
includes all the abelian groups of order pa+l that are contained in G and include H. Hence the abelian subgroups of