(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 -> 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 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 FD2n (D-2n=< a,b|a(n)=b(2)=1,b(-1)ab=a(-1)>, n >= 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.