Free products in the unit group of the integral group ring of a finite group

TL;DR

This paper constructs explicit free products and monoids within the unit group of the integral group ring of finite groups, advancing understanding of their algebraic structure and providing concrete examples for nilpotent and arbitrary finite groups.

Contribution

It provides explicit generic constructions of free products and monoids in the unit group of integral group rings for finite groups, including nilpotent groups, extending previous results.

Findings

Constructed free products of cyclic groups of prime order in unit groups.

Generated free products of p^k and p^m cyclic groups for nilpotent groups of class 2.

Established generic methods for creating free monoids in unit groups of finite groups.

Abstract

Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and free monoids in the unit group $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$. For a nilpotent group $G$ with a non-central element $g$ of order $p$, explicit generic constructions are given of two periodic units $b_1$ and $b_2$ in $\mathcal{U}(\mathbb{Z}G)$ such that $\langle b_1 , b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb{Z}_p \star \mathbb{Z}_{p}$, a free product of two cyclic groups of prime order. Moreover, if $G$ is nilpotent of class $2$ and $g$ has order $p^n$, then also concrete generators for free products $\mathbb{Z}_{p^k} \star \mathbb{Z}_{p^m}$ are constructed (with $1\leq k,m\leq n $). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gon{\c{c}}alves-Passman. Further, for an arbitrary finite group $G$ we give generic constructions of free monoids in $\mathcal{U}(\mathbb{Z}G)$ that generate an infinite solvable subgroup.