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

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INVARIANTS OF ABELIAN GROUPS

89

the elements of G have been exhausted, it has been proved that
every non-cyclic abelian group of order pm is the direct product of
independent cyclic groups. This is the most important theorem
relating to abelian groups.*
These independent cyclic groups may be represented as
substitution groups on distinct sets of letters. Moreover,
it is clear that a group can be constructed such that the orders
of such substitution groups are arbitrary, and hence the product is an abelian group of arbitrary order. That is, the number
of distinct abelian groups of order pm is equal to the number of the
possible partitions of m as regards addition, and each of these
groups may be completely defined by the value of p and the
symbol (ml, m2,.., mx) where mi, m2,..., mx represent positive integers such that ml+m2+.. +.. mx=m. The
group G is completely defined by p and the values of the integers mi, m2,..., mx, and it is not affected by the order in
which these integers are arranged. It is said to be of order pm
and of type (ml, m2,..., mx). We may therefore suppose that
the numbers mi, m2,..., mx are always arranged in order
of magnitude, beginning with the largest. It may be added,that the determination of the number of possible abelian groups
of order pm is reduced by these theorems to a problem in the
theory of numbers; viz., the determination of the total number of possible partitions of m as regards addition. This problem has received considerable attention, but it still involves
many unsolved difficulties.
Suppose that an abelian group G of order pm has a invariants.
All of its elements whose orders divide p must constitute a subgroup of order pa; and, conversely, whenever these elements
constitute a group of order p", G has exactly a invariants. The
* A set of independent generating elements can generally be selected in a
large number of ways. Such a set is often called a base of the abelian group,
and the operators si, S2, Ss,... are called elements of the base. The fundamental theorem that every abelian group is a direct product of independent
cyclic groups is implicitly contained in the works of C. F. Gauss and E. Schering,
but L. Kronecker gave the first satisfactory proof of it in 1870. It should perhaps
be placed next after Cayley's theorem among the most fundamental theorems
of group theory.