Derivations of group rings for finite and FC groups

TL;DR

This paper develops decomposition theorems and algorithms for derivations of group rings, especially for finite and FC groups, providing explicit calculations and examples.

Contribution

It introduces a topological technique and explicit algorithms for derivations of group rings for finite and FC groups, expanding understanding and computational methods.

Findings

Describes all derivations of $A[G]$ for finite and FC groups.

Provides an algorithm for explicit derivation calculation.

Includes examples with specific group rings.

Abstract

In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique for studying derivations of a group ring $A[G]$ in case $G$ has finite conjugacy classes. As a result, we describe all derivations of algebra $A[G]$ for the case when $G$ is a finite group, or $G$ is an FC-group. In addition, we describe an algorithm to explicitly calculate all derivations of a group ring $A[G]$ in case $G$ is finite. As examples, derivations of $\mathbb{Z}_4[S_3]$ and $\mathbb{F}_{2^m}D_{2n}$ are considered.