Postulates for the inverse operations in a group

It may happen that the function F is of such a character that when the 5-symbols z, x are given in (1), an c5-symbol y is uniquely determined, and when the25-symbols y, z are given in (1), an25-symbol x is uniquely determined. In this case we may associate with the function F(x, y) two other one-valued functions y = G (z, x), x = H (y, z) defined over the collection (B. These functions are called the first and second inverses of the function F. The primary object of this paper is to state the restrictions which must be imposed upon F in order that one of its inverses may define with (E an abstract group. It is convenient, in developing the properties of the system { ( ;o} consisting of the i-symbols, F, and one or more postulates, to replace (1) by the notationt z = xoy.