The associative algebra of derivations of a group algebra

Abstract

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group [Formula: see text] over a finite field [Formula: see text] are determined such that the set of derivations of the group algebra form an associative [Formula: see text]-algebra. The derivations of [Formula: see text] form a nontrivial associative [Formula: see text]-algebra if and only if [Formula: see text] has characteristic 2 and [Formula: see text] is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting [Formula: see text]-algebra is determined.