(σ,τ)-derivations of group rings with applications

Leo Creedon and Kieran Hughes in [18] studied derivations of a group ring RG (of a group G over a commutative unital ring R) in terms of generators and relators of group G. In this article, we do that for $(\sigma ,\tau )$-derivations. We develop a necessary and sufficient condition such that a map $f:X\to RG$ can be extended uniquely to a $(\sigma ,\tau )$-derivation D of RG, where R is a commutative ring with unity, G is a group having a presentation $〈X|Y〉$ (X the set of generators and Y the set of relators) and $(\sigma ,\tau )$ is a pair of R-algebra endomorphisms of RG which are R-linear extensions of the group endomorphisms of G. Further, we classify all inner $(\sigma ,\tau )$-derivations of the group algebra RG of an arbitrary group G over an arbitrary commutative unital ring R in terms of the rank and a basis of the corresponding R-module consisting of all inner $(\sigma ,\tau )$-derivations of RG. We obtain several corollaries, particularly when G is a $(\sigma ,\tau )$-FC group or a finite group G and when R is a field. We also prove that if R is a unital ring and G is a group whose order is invertible in R, then every $(\sigma ,\tau )$-derivation of RG is inner. We apply the results obtained above to study σ-derivations of commutative group algebras over a field of positive characteristic and to classify all inner and outer σ-derivations of dihedral group algebras $F{D}_{2n}$ ($D_{2n}=〈a,b|a^n=b^2=1,b^{-1}ab=a^{-1}〉$, $n\ge 3$) over an arbitrary field $F$ of any characteristic. Finally, we give the applications of these twisted derivations in coding theory by giving a formal construction with examples of a new code called IDD code.