Computing generators of the unit group of an integral abelian group ring

TL;DR

This paper presents an algorithm to compute generators of the unit group of integral group rings for finite abelian groups, with implementation results up to order 110, and extends to more general algebraic groups.

Contribution

It introduces a new algorithm for generating units in integral abelian group rings and provides practical implementation and results, including the index of Hoechsmann units.

Findings

Successfully computed unit groups for groups of order up to 110

Determined the index of Hoechsmann units in the full unit group

Extended the algorithm to diagonalizable algebraic groups

Abstract

We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110. In particular for those cases we obtained the index of the group of Hoechsmann units in the full unit group. At the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalizable algebraic group.