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

100

ABELIAN GROUPS

[CH. IV

While it is not difficult to find, by means of the theorem
stated above, the total number of the. different types of subgroups in a given abelian group whose invariants are known, it
is a problem of considerable difficulty to determine all the possible subgroups of the same type. To illustrate this fact we
consider the subgroups of the important class of abelian groups
of order pm and of type (1, 1,..., to m units). In this case
there are evidently m-1 different types of subgroups, excluding the identity. That is, there is one and only one type of
subgroups of order pa, a= 1 2,..., m-1, separately.
In this case it is also not very difficult to determine the
number of the different subgroups of order pa. In fact, this
number is clearly equal to the quotient obtained by dividing
the total number of ways in which generating operators of such
a subgroup can be selected from the operators of the group
by the number of ways in which such generators can be selected
from the operators of the subgroup. Hence this number is
(pm-1)(p__p)(p )... (pm pa-1)
(pa- (pa -p) (pa - P) * * (p a-pa1)
(p_ l)(p-1 )... (po-a+l- )
(pa_)(pa-l_1)... (p-l)
In the particular case when a=m-l, this formula reduces to
p"m-
p-1
Hence there are as many subgroups of order pm-l in an abelian
group of order pm and of type (1, 1, 1,... ), as there are subgroups of order p. For instance, the group of order 8 and of
type (1, 1, 1) has seven subgroups of order 2 and also seven
subgroups of order 4, while the group of order 16 and of type
(1, 1, 1, 1) has fifteen subgroups of order 2 and fifteen subgroups
of order 8.