The group determinant determines the group

Abstract

Let G = { g 1 , … , g n } G = \left \{ {{g_1}, \ldots ,{g_n}} \right \} be a finite group of order n n , let K K be a field whose characteristic is prime to n n , and let { x g | g ∈ G } \left \{ {{x_g}\left | {g \in G} \right .} \right \} be independent commuting variables over K K . The group determinant of G G is the determinant of the n × n n \times n matrix ( x g i g j − 1 ) \left ( {{x_{{g_i}g_j^{ - 1}}}} \right ) . We show that two groups with the same group determinant are isomorphic.