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

160

ISOMZORPHISMS

[CH. VII

In its most elementary form the concept of isomorphisms
is one of the oldest in mathematics, as it lies at the base of the
development of abstract numbers. The concept of abstract
number evidently rests on a kind of isomorphism between concrete units of various kinds, so that for many purposes we
may fix our attention entirely on what is common, viz., the
abstract concept of units. In the theory of groups the concept of isomorphisms assumes a new importance in view of the
fact that the different automorphisms of a group may be represented by the corresponding substitutions on its operators, and,
as was noted in ~ 19, the totality of these substitutions constitute a group known as the group of isomorphisms,* or the
group of automorphisms, of the original group. We have
thus associated with each group its group of isomorphisms,
which is of fundamental importance in many applications of
the group.
62. Group of Isomorphisms as a Substitution Group. If
g distinct letters are placed in a (1, 1) correspondence with the
operators of the group G of order g, the symmetric substitution
group will correspond to the totality of the possible arrangements of the operators of G. Such an arrangement cannot
correspond to an automorphism of G unless the identity corresponds to itself. Hence the group of isomorphisms I of G can
always be represented as a substitution group on at most g -1
letters, and its order must therefore be a divisor of (g-l)!.
This order cannot be equal to (g-1)! except in case of three
groups besides the identity, viz., the groups of orders 2 and 3,
and the four-group. In fact, it is evident that I cannot be more
than doubly transitive on g-1 letters, since the correspondence
of two operators fixes the correspondence of their product.
In particular, the order of the group of isomorphisms of any
finite group is a finite number.
A necessary condition that I be transitive on g-1 letters
is that all the operators of G besides the identity have the same
* Isomorphisms were first studied in an explicit manner by C. Jordan and
A. Capelli. Their group properties were first studied by 0. Holder and by E. H.
Moore.