The Group Determinant Determines the Group

Abstract

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