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

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SPECIAL CLASS OF ABELIAN GROUPS

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This group could have been equally well represented by
1 13.57 15-37 17-35
and it is clear that it is simply isomorphic with the group formed
by the following numbers, when they are combined by multiplication:
1 3 5 7 (mod 8).
These examples may suffice to illustrate the fact that the
group formed by the 0(m) totitives, with respect to multiplication (mod m), is the group of isomorphisms of the cyclic group
of order m. To prove this fact it is only necessary to observe
that the correspondence of the operators of lower orders in a
cyclic group is completely determined by the correspondence
of the operators of highest order, and that all of the latter may
be obtained from any one of them by raising it to all the various
powers which are prime to the order of the cyclic group. In
particular, a necessary and sufficient condition that the number m has a primitive root is that the group formed by the
q(m) totitives (mod m) be cyclic. While the group formed by
the totitives is always abelian, there are many abelian groups
which cannot be represented in this way. Hence these groups
form a special class of abelian groups.
We proceed to determine some conditions which must be
satisfied in order that an abelian group G may belong to this
class. When G is cyclic the matter is quite simple. It is
necessary and sufficient that its order g be the exponent to
which a primitive root of some number belongs. That is,
whenever g=pa(p-1),* p being an odd prime number, and a
being any positive integer or zero, the cyclic group G belongs
to the given class, and only then. The lowest two even numbers which are not of the form pa(p-1) are 8 and 14; hence
these numbers are the lowest orders of cyclic groups of an
even order that cannot be the groups of isomorphisms of any
cyclic groups whatever, and hence the cyclic groups of these
two orders cannot be represented as groups of totitives.
* Cf. Dirichlet-Dedekind, Zahlentheorie, 1879, p. 340.