Algebras defined by finite groups

Introduction. 1. In a paper presented to this Society at a previous meeting, t the general structure of linear associative algebra was discussed and certain fundamental theorems of great generality were proved. The present paper is an application of these theorems to the study of what may be called group algebras. By a group algebra is meant that linear algebra whose units are defined to be such that each unit et corresponds to an operator Oi of some given abstract finite group, f and conversely, and such that for each equation of the group 0 0. = 0, corresponds an equation eze. = ek of the algebra. From the syinbolic point of view the algebra differs from the group only in that expressions-for brevity, let us say, numbers-of the form Ex. e. are possible, wherein the coefficients x, are any scalars. ? That this algebra is linear and associative, is obvious from the definition. When no confusion is feared, the notations and terminology of the group and of the algebra will be used interchangeably. 2. In the paper cited as Theory is developed the theorem that the numbers of a linear associative algebra are subject to the laws of matrices, and certain conclusions are drawn from this fact. This method of development enables us to make immediate use of any theorem needed which is true of matrices, and so saves a redevelopment of many such theorems. In the present paper I shall consider the numbers as multiple algebraic entities, referring to the theory of matrices only when some needed theorem is to be translated into an algebraic theorem. 3. In Part 1 of the paper the general form of any grotup algebra is to be dis. cussed and certain general theorems established. In Part 2 a few special cases