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

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 $⟨b_1,b_2⟩=⟨b_1⟩\star ⟨b_2⟩\cong {\mathbb{Z}}_p\star {\mathbb{Z}}_p$, 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 $1\le k,m\le n$). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gon{ç}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.

10 pages

References in corpus (2)

• On uniform exponential growth for solvable groups
• Free subgroups in group rings