Derivations on Group Algebras with Coding Theory Applications

TL;DR

This paper classifies derivations of group algebras based on group generators and relations, explores their triviality in certain cases, and applies these findings to construct binary codes.

Contribution

It provides a comprehensive classification of derivations for various group algebras and demonstrates their application in coding theory.

Findings

Derivations are trivial for semisimple group algebras of abelian groups.

Explicit derivations are constructed for finite group algebras, including dihedral groups.

Applications include constructing binary codes from derivations of group algebras.

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.