12.92 (1992)

For a field $K$ of characteristic 2, each finite group $G = \{g_1, \dots, g_n\}$ of odd order $n$ is determined up to isomorphism by its group determinant (Formanek–Sibley, 1990) and even by its reduced norm which is defined as the last coefficient $s_m(x)$ of the minimal polynomial $\phi(\lambda; x) = \lambda^m - s_1(x)\lambda^{m-1} + \dots + (-1)^m s_m(x)$ for the generic element $x = x_1 g_1 + \dots + x_n g_n$ of the group ring $KG$ (Hoehnke, 1991). Is it possible in this theorem to replace $s_m(x)$ by some other coefficients $s_i(x)$, $i < m$? For notation see (G. Frobenius, Sitzungsber. Preuss. Akad. Wiss. Berlin, 1896, 1343–1382) and (K. W. Johnson, Math. Proc. Cambridge Phil. Soc., 109 (1991), 299–311).