Free subgroups in group rings

TL;DR

This paper introduces a straightforward method for constructing free subgroups within the unit group of a group ring, demonstrating the existence of free products of cyclic groups and specific subgroup structures.

Contribution

It provides a new simple approach to identify free subgroups in the unit group of group rings, including explicit constructions of free products and cyclic extensions.

Findings

V(KG) always contains the free product C_n*C_n of two finite cyclic groups

Constructs examples of subgroups that are cyclic extensions of non-abelian free groups

Demonstrates the presence of free objects in the unit group of group rings

Abstract

Let V(KG) be the normalized group of units of the group ring KG of a non-Dedekind group G with nontrivial torsion part t(G) over the integral domain K. We give a simple method for constructing free objects in V(KG).In particular, we show that V(KG) always contains the free product C_n*C_n of two finite cyclic groups. We construct examples of subgroups in V(KG), which are either cyclic extensions of a non-abelian free group or C_n*C_n.