Nilpotent elements in representation rings over fields of characteristic 2

Abstract

whereg(m an) =gm @gn,gEG, mEM, neN. If p is an odd prime, it has been shown that a(kG) is semisimple if and only if thep-Sylow subgroups of G are cyclic (Green [3], O’Reilly [4], Zemanek [S]). I f p = 2, a(kG) is semisimple if the 2-Sylow subgroups of G are cyclic [3,4]. However, the converse is false. Conlon [l] and Wallis [7] have shown that a(kG) is semisimple if the 2-Sylow subgroup is a (2,2)-group. The question being settled for 2-groups of order at most 4, we were led to consider the noncyclic groups of order 8. In this paper we shall prove the following: