$(σ, τ)$-Derivations of Group Algebras

Abstract

In this article, a classification of all the (σ, τ )-derivations of group algebras is given in terms of the generators and relators of a group. If R is a commutative ring and G = X | Y is a group with X as its set of generators and Y as a set of relators, then a necessary and sufficient condition is developed for a map f : X → RG to be uniquely extendable to a (σ, τ )-derivation D of RG, where (σ, τ ) is a pair of endomorphisms of RG which are R-linear extensions of the group homomorphisms of G. Amongst several results established as applications of the above proved results, a classification of all the σ-derivations of commutative group algebras over a field of positive characteristic is given.