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

CHAPTER III

FUNDAMENTAL DEFINITIONS AND THEOREMS OF
ABSTRACT GROUPS
21. Introduction. In the preceding chapters we gave
several examples of groups which were either in the form of
substitution groups or could readily be associated with such
groups. Some of the fundamental theorems of substitution
groups were also developed. As the theory of groups is applicable to many different subjects, it became a matter of economy
of thought to develop and to state the main theorems of this
theory in a language which is common to these various subjects. The efforts to accomplish this end led gradually to
what is known as abstract group theory.
Many of the theorems proved in the preceding chapter
can be used directly in the theory of abstract groups. In fact,
we shall find that the theory of substitutions is a very useful
means to study the abstract properties of groups. It has
already been observed, ~ 15, that the question of primitivity and
imprimitivity of a substitution group has an abstract meaning,
even if these terms apparently relate only to the notation of
substitution groups.
Some of the definitions used in the theory of substitution
groups are not directly available for the study of abstract
groups, and hence it will be necessary to re-define some of the
terms used in the preceding chapter. This is due to the fact
that substitutions have inherent properties. For instance,
the associative law is always satisfied when substitutions are
multiplied, and hence we did not need to specify this law in
defining a substitution group. On the contrary, this law
should be included in a definition of an abstract group.
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