Modular Theory of Group - Matrices

1. The importance of the concept group-matrix in the theory of finite groups was recognized by Dedekind as early as 1880. The development of a general theory of group-matrices is due, however, to Frobenius, (Berliner Sitzungsberichte, from 1896 to the present). In particular, Frobenius has applied the theory to the representation of a finite group as a (non-modular) linear group. Since linear congruence groups are of importance in the theory of groups, particularly in questions of isomorphisms, and play a fundamental rôle in the applications of groups to the theory of functions and to geometry, the study of the representations of a finite group as a linear congruence group is of decided importance. It is here proved that, if p" is the highest power of the prime p dividing the order a of a group G, the group-matrix of G can be transformed, by a matrix whose elements are integers taken modulo p, into a compound matrix in which the submatrices to the right of the main diagonal have zero elements throughout, while the p" submatrices in the diagonal are identical. Let D denote one of the diagonal submatrices, so that D is a square matrix of order g/])". Then the group-determinant A of G is congruent to | D \p" modulo p. This result is in marked contrast to the non-modular theory, in which each algebraically irreducible factor of A enters to a power exactly equal to its degree. It is shown in § 8 that the group-matrices of all groups of order p* can be reduced to their canonical form modulo p by one and the same transformation. An interesting theorem on group-characters is obtained in § 11. Just as simplicity is attained in the algebraic theory only when certain algebraic irrationalities are introduced to permit of the complete factoring of the group-determinant into algebraically irreducible factors, so corresponding simplicity in the modular theory can be attained only by the use of Galois imaginaries (roots of irreducible congruences) in order to normalize completely the diagonal matrices D (§ 10). It is therefore proposed to take as the field of reference the field F defined as the aggregate of the Galois fields GF\_pn"\ ,