The associative algebra of derivations of a group algebra

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group G over a finite field K are determined such that the set of derivations of the group algebra form an associative K-algebra. The derivations of KG form a nontrivial associative K-algebra if and only if K has characteristic 2 and G 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 K-algebra is determined.