Describing units of integral group rings up to commensurability

Florian Eisele,Ann Kiefer,Inneke Van Gelder

doi:10.1016/j.jpaa.2014.09.031

Florian Eisele, Ann Kiefer + Show 1 more

Open Access

https://doi.org/10.1016/j.jpaa.2014.09.031

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Abstract

We restrict the types of 2×2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring ZG for all finite groups G that do not have a non-commutative Frobenius complement as a quotient.

Highlights

For a group G we denote by U(ZG) the unit group of the integral group ring ZG
For several classes of finite groups G including nilpotent groups of odd order, Ritter and Sehgal [2, 3, 4] showed that the Bass units together with the bicyclic units generate a subgroup of finite index in U(ZG)
Jespers and Leal [5] proved that the group generated by the Bass units and the bicyclic units is of finite index in U(ZG) for finite groups G which do not have any non-abelian homomorphic images that are Frobenius complements and, whose rational group ring QG does not have any simple components of one of the following types: 1. a 2 × 2-matrix ring over the rationals; 2. a 2 × 2-matrix ring over a quadratic imaginary extension of the rationals; 3. a 2 × 2-matrix ring over a non-commutative division algebra

Summary

Introduction

For a group G we denote by U(ZG) the unit group of the integral group ring ZG. Bass [1] proved that if C is a finite cyclic group, the so-called Bass cyclic units generate a subgroup of finite index in U(ZC). Jespers and Leal [5] proved that the group generated by the Bass units and the bicyclic units is of finite index in U(ZG) for finite groups G which do not have any non-abelian homomorphic images that are Frobenius complements (equivalently, are fixed point free, see Definitions 2.4 and 2.5) and, whose rational group ring QG does not have any simple components of one of the following types: 1. Matrix ring over a quadratic imaginary extension of the rationals or a 2 × 2-matrix ring over a totally definite quaternion algebra over Q, this component is a 2 × 2-matrix ring over one of the following rings: All of these fields respectively skew-fields contain a norm Euclidean maximal order (see [9, Proposition 6.4.1] and [10, Theorem 2.1]). Other constructions of subgroups of finite index of U(ZG) for some specific classes of finite groups G, using a description of the center Z(U(ZG)) and a description of the matrix units in QG, have been given for example in [11, 12]

Preliminaries

Main results

A is simple and neither a skew-field nor a field

A is a skew-field or a field

Examples