A definition of abstract groups

Dr. E. V. HUNTINGTON has recently given two different definitions of abstract groups by sets of respectively three t and four ilndependent postulates; these definitions were perhaps suggested by those given by WEBER and by BURNSIDE. Some years ago Professor J. PIERPONT and I independently hit on a type of definition very desirable from the grouptheoretic stanidpoint. In ? 1 I formulate a definiition of this type by means of five independent postulates; the independence is proved in ? 4. In ? 3 the definition is related to other definitionis. In this connection I am led to a slightly modified definition (p. 489) by means of six independent postulates, which, even from the standpoint of abstract logic, seems to me simpler than either of those of Dr. HUNTINGTON. ? 1. THE DEFINITION.