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 $\langle b_1, b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb {Z}_p \star \mathbb {Z}_{p}$, 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 Gonçalves-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.