224 Mr. Gr. A. Miller on the Groups of Isomorphisms 
The theory of the groups of isomorphisms of cyclic groups 
includes the theory of primitive roots in number theory in 
view of the known theorem that the necessary and sufficient 
condition that a number (m) has primitive roots is that the I 
of the cyclic group of order m is cyclic. It has also been 
observed that some of the most useful theorems of number 
theory, such as Fermat's and Wilson's, are included in 
elementary theorems relating to the group of isomorphisms *. 
Moreover, the groups of isomorphisms furnish one of the 
best means to construct groups having known subgroups, 
and in many other group-theory considerations they play 
a fundamental role which is continually receiving more 
attention. 
§ 1. General Theorems. 
Theorem I. — If an abelian group G which involves operators- 
whose orders exceed 2 is extended by means of an operator of 
order 2 which transforms each operator of G into its inverse y 
then the group of isomorphisms of this extended group is the 
holomorph of Gr. 
As the non-invariant operators of order 2 in the extended 
group correspond to themselves in every holomorphism of 
this group, its I may be represented as a transitive substi- 
tution group of degree n, where n is the order of G. After 
any holomorphism of G has been established, these operators 
of order 2 may be arranged in n different ways. Hence the 
order of I is n times the order of the I of G. Moreover, I 
contains an invariant subgroup which is simply isomorphic 
with G since we may obtain a holomorphism of the extended 
group by multiplying each operator of G by the identity 
and the remaining operators of the extended group by an 
arbitrary operator of G. It may be observed that this theorem 
includes the known theorem that the I of the dihedral group 
of order 2m, m > 2, is the holomorph of the cyclic group of 
order m. 
Theorem II. — The square t of a complete group which is 
not a direct product has the double holomorph of this group for 
its group of isomorphisms. 
Let H and H' be two identical complete groups which are 
not direct products, and let G be the direct product of H 
* Cf. Transactions of the American Mathematical Society, vol. iv. 
1903) p. 158 ; < Annals of Mathematics,' vol. iv. (1903) p. 188. 
f The direct product of two identical groups is called the square of 
one of them. 
