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

~ 41] ISOMORPHISMS OF AN ABELIAN GROUP

101

EXERCISES
1. A necessary and sufficient condition that a group be abelian is that
each operator corresponds to its inverse in one of the possible automorphisms of the group.*
2. Find the number of subgroups with invariants 6, 2 in the abelian
group whose invariants are 12, 6, 2.
3. Determine the number of the subgroups of each possible order
in all the abelian groups of order p3, p being a prime.
4. Every abelian group is generated by its operators of highest order.
5. Give an instance of a non-abelian group which is not generated by
its operators of highest order.
41. Group of Isomorphisms of an Abelian Group.t Some
of the most useful properties of an abelian group are exhibited
by its group of isomorphisms. We have already considered the
group of isomorphisms of a cyclic group and found that it is
an abelian group. We shall see that a necessary and sufficient
condition that the group of isomorphisms of an abelian group
G be abelian is that G be cyclic, and hence it results that the
groups of totitives are the only abelian groups of isomorphisms
of abelian groups. This fact is a special case of the theorem
that the invariant operators of the group of isomorphisms of any
abelian group constitute a group which is simply isomorphic
with the group of the totitives of the largest possible invariant
of this abelian group. For instance, if an abelian group has the
invariants 10, 10, 2, the invariant operators of its group of
isomorphisms constitute the cyclic group of order 4. We proceed to prove the stated theorem.
We shall first prove that if an operator t of the group of
isomorphisms of an abelian group G transforms every operator
of G into the same power (rth) of itself it must be commutative
with every operator of the group of isomorphisms of G. Let
ti be any other operator of this group of isomorphisms and suppose that
tl 1satl =iS t- st = S,
* This includes the theorem that every group which involves no operator
whose order exceeds 2 is abelian.
t Cf. A. Ranum, Transactions of the American Mathematical Society, vol.
8 (1907), p. 83.