Derivations on group algebras with coding theory applications

Abstract

This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary and sufficient conditions on a map from S to RG are established, such that the map can be extended to an R-derivation of RG. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of finite group algebras are constructed and listed in the commutative case and in the case of dihedral groups. In the dihedral case, the inner derivations are also classified. Lastly, these results are applied to construct well known binary codes as images of derivations of group algebras.

Derivations of group rings for finite and FC groups

“…The last example is important for applications especially for cryptography. So we will give a new combinatorial way to get such results from [19,17,18].…”

“…Moreover, in recent paper [7] character technique proved to be useful for studying (σ, τ )-derivations. Derivations, and especially (σ, τ )-derivations of group rings have various applications in coding theory [12,19].…”

“…In our paper we prove the decomposition theorems which provide the way to explicitly describe all derivations of a finite group ring. This can be useful for coding theory applications (see [12,19]). The decomposition theorems we prove also give rise to the criterion for solving the problem of outer derivations' existence for finite group rings.…”

“…The main goal of this section is to reproduce the result of paper [19]. In order to calculate derivations of F 2 m D 2n , we use several properties of dihedral groups, proofs of which can be found in expository papers by Keith Conrad [17,18].…”

“…Our method opens the way for practical applications of derivation algebras. As an example, the explicit description of derivation algebras is useful for construction of binary codes (see [19]).…”