Definition of improper groups by means of axioms : a dissertation

Abstract

Essentially, a group is an associative field, in which the inverse combinations are uniquely possible. This is a concise statement of the classical definition of a group. The conditions which it connotes will be used here as a convenient starting point for comparing various definitions of a group. It should be said, at the outset, that the accurate definition of a group will be somewhat different question according as a finite or an infinite, or an unspecified number of objects is to be dealt with. Conditions which are sufficient in one case may be found insufficient in the others.