Postulates for Special Types of Groups

In defining a special type of group, such as commutative group or finite group, one may be content to add a suitable postulate, or possibly a number of postulates, describing the special property under consideration, to any set of postulates for a general group. I t is, however, of interest to pursue the matter further, to determine whether a simplified set of postulates can be set up which will adequately describe the special type. A number of investigations of this sort have been made; the results of Weber, Huntington, and Hurwitz are especially worthy of note, f In the present paper I shall use as a basis the three-postulate definition of group which I recently presented in this Bulletin.X Let there be given a set of elements G(a, b, c, • • • ) and a rule of combination, which may be called multiplication, by which any two elements, whether they be the same or different, taken in a specified order, determine a unique result which may or may not be an element of G. This system forms a group if it satisfies the following three postulates : I. If a, b, c, ab, be, (ab)c, a(bc) are all elements of G, then (ab)c = a(bc). II . If a and b are elements of G, there exists an element x of G such that ax — b. I I I . If a and b are elements of G, there exists an element y of G such that ya = b. The reader will recall that a familiar four-postulate definition of group employs these three postulates, and a closure postulate. That definition was due to Huntington and Moore and was, in