Central units in some integral group rings

Abstract

Let G be a finite group and $\mathbb{Z}G$ be the integral group ring of G. We denote by $U_1(\mathbb{Z}G)$ the group of normalized units of $\mathbb{Z}G;$ that is, the units which have augmentation 1, and by $Z(U_1(\mathbb{Z}G))$ the group of normalized central units. Many articles have been written describing the groups $U_1(\mathbb{Z}G)$ and $Z(U_1(\mathbb{Z}G))$ for certain groups G. In this work, we will describe the group of normalized central units of some integral group rings by applying the idea presented in an article by Ferraz and Simón to a wider variety of groups, and we will study some examples of groups that can be treated with this method: metacyclic groups of type $C_{q^m}⋊C_{p^n};$ some metacyclic p-groups; some metabelian p-groups and some generalized dihedral groups.

Keywords:

• Central units
• integral group rings
• metabelian groups

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C05

Additional information

Funding

The first author was partially supported by CNPq.