$(\sigma, \tau)$-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 $f:X \rightarrow RG$ 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 $\langle X \mid Y \rangle$ ($X$ 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 $\sigma$-derivations of commutative group algebras over a field of positive characteristic and to classify all inner and outer $\sigma$-derivations of dihedral group algebras $\mathbb{F}D_{2n}$ ($D_{2n} = \langle a, b \mid a^{n} = b^{2} = 1, b^{-1}ab = a^{-1}\rangle$, $n \geq 3$) over an arbitrary field $\mathbb{F}$ 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.