On the group determinant

Abstract

The original motivation for the introduction by Frobenius of group characters for non-abelian groups was the problem of the factorization of the group determinant corresponding to a finite group G. The original papers are [5] and [6] and a good historical survey of the work is given in [7] and [8]. If G is of order n, the group matrix XG is defined to be the n×n matrix {xg, h} where xg, h = xgh∈G. Here the xg, g∈G, represent variables. The group determinant ΘG is defined to be det(XG), and is thus a polynomial of degree n in the xg. This determinant is the same, up to sign, as that of the matrix obtained from the unbordered multiplication table of G by replacing each element g by xg. If there is no ambiguity ΘG will be written as Θ.