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

~ 62] ISOMORPHISMS REPRESENTED BY SUBSTITUTIONS 161

order, and hence g must be of the form p", p being a prime
number. Since the correspondence of two operators determines
the correspondence of their powers, it is clear that I cannot be
a primitive substitution group unless p=2. If all the operators of G besides the identity are. of order 2, 1 is evidently
doubly transitive. Hence it results that a necessary and sufficient condition that I be primitive on g-1 letters is that all
the operators of G be of order 2.*
The group I is generally intransitive on g-1 letters, and the
number of its systems of intransitivity is equal to the number
of complete sets of conjugate operators of G under I. In particular, the number of characteristic operators of G is equal to
g diminished by the degree of 1. A sufficient condition that I
is simply isomorphic with one of its transitive constituents,
when it is represented as such a substitution group, is that
G is generated by one of its complete sets of conjugates under I.
When G is abelian it is generated by its operators of highest
order, and these constitute a complete set of conjugates under I.
As they constitute the only complete set of such conjugates
that generate G, it results that the group of isomorphisms of an
abelian group can always be represented in one and in only one
way as a transitive substitution group on letters corresponding
to operators of this abelian group. In other words, if the group
of isomorphisms of an abelian group is represented on letters
corresponding to the various operators of this abelian group,
this group of isomorphisms has only one transitive constituent
which is simply isomorphic with it, since every abelian group
of order pm, except the group of order 2, admits a non-identical
isomorphism in which every operator which is not of highest
order corresponds to itself. A like theorem does not apply
in general to the non-abelian groups. In fact, if the I of a nonabelian group is represented on the g-1 letters corresponding
to the operators of the group, excepting the identity, the
number of its transitive constituents which are simply isomorphic
with I may vary from zero to an indefinitely large number, as
results from the alternating groups.
* E. H. Moore, Bulletin of the American Mathematical Society, vol. 2 (1896), p. 33.