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

~ 41] ISOMORPHISMS OF AN ABELIAN GROUP

103

dition that an operator of the group of isomorphisms of an abelian
group be invariant under this group, is that it should transform
every operator of this abelian group into the same power of itself.*
Since two abelian groups having the same invariants can be
made simply isomorphic, and two simply isomorphic groups
have the same invariants, it results that the order of the group
of isomorphisms expresses also the number of different ways
of choosing the independent generators of the group. It
should be observed that while every operator of highest order
in an abelian group may be used as an independent generator,
and hence each operator of highest order must correspond to
every other operator of this order in some simple isomorphism
of the group with itself, it is not generally true that every
operator of lower order corresponds to every operator of its
own order in some simple isomorphism of the group.
This fact may be illustrated by means of the abelian group
whose invariants are p2 and p. It is evident that this group
contains a characteristic subgroup of order p; viz., the subgroup
of order p which is generated by its operators of order p2.
The remaining p subgroups of order p in the given group of
order p3 are conjugate under the group of isomorphisms of this
group.
In any automorphism of any abelian group G each operator
of G corresponds to itself multiplied by some operator of G.
The totality of these multiplying operators evidently constitutes a group T which is either G itself or a subgroup of G,
and the automorphism may be obtained by making G isomorphic
with T and multiplying corresponding operators. In this
isomorphism no operator except the identity can correspond
to its inverse. As this condition is necessary as well as sufficient
we have arrived at the following fundamental
THEOREM: Every automorphism of an abelian group G may
be obtained by (1) making G isomorphic with one of its subgroups
or with itself in such a manner that no operator besides the identity
corresponds to its inverse, and (2) making each operator of G
* Transactions of the American Mathematical Society, vol. 1 (1900), p. 397;
vol. 2 (1901), p. 260.