On the Direct Product in the Theory of Finite Groups

1. THIs paper is an inrvestigation into the formation of groups by the process known as the "direct product." The calculus introduced by Frobenius is used throughout with the addition of the sign A -, B to denote the largest sub-complex which is common to the complexes A and B; simple isomorphism between A and B is denoted by A B. Groups are represented by Gothic letters and their elements by the corresponding Roman capitals; ordinary numbers are represented by small Roman letters, and in particular the order of a group is denoted by the same letter of the alphabet as the group itself. Thus the elements of a group A are denoted by Al, Aj, A. , AHa a being its order. The identity is denoted as usual by .