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

~ 40],

SUBGROUPS AND QUOTIENT GROUPS

99

5. If the group of totitives of m has for its order a power of a prime
this order is of the form 2'.
6. Every possible abelian group is a subgroup of some group of
totitives.
Suggestion: If the invariants of the given abelian group are so chosen
that each is a power of a prime number, it is clearly possible to choose
m so that the group of totitives of m involves the same invariants.
40. Subgroups and Quotient Groups of any Abelian Group.
It has been proved that every abelian group may be regarded
as the direct product of cyclic groups and hence it is completely
determined by the orders of these groups. As every subgroup
of an abelian group is abelian, it results that these subgroups
are also completely determined by the orders of the cyclic
groups of which they are the direct products. Hence it follows
immediately that a necessary and sufficient condition that an
abelian group G whose invariants are ii, i2,..., ip contains
a subgroup whose invariants are jl, j2,..., j is that it be
possible to associate the tj's with t distinct i's so that each i is
equal to or greater than the corresponding j.
If such an arrangement were not possible the subgroup
would involve more operators of a certain order than the entire
group. The condition imposed upon the invariants of a subgroup is clearly equally applicable as regards the invariants
of a quotient group. Hence we have the important theorem:
The invariants of any subgroup of an abelian group are invariants of a quotient group, and the invariants of any quotient group
are also invariants of a subgroup. In other words, each subgroup
is simply isomorphic with a quotient group and vice versa.
A like theorem is not always true as regards non-abelian groups.
If a group is cyclic all of its subgroups may be obtained by
raising successively all of its operators to the same power, but
this method cannot give all the subgroups of a non-cyclic
group. The kth power of each operator of an abelian group G
gives a group which is simply isomorphic with G whenever k
is prime to g. If k is not prime to g, these kth powers constitute a quotient group of G, whose order is g divided by the
total number of the operators of G whose orders divide k.