Embedding free products in the unit group of an integral group ring. (Q1881513)

\textit{B. Hartley} and \textit{P. F. Pickel} [Can. J. Math. 32, 1342-1352 (1980; Zbl 0458.16007)] showed that for a finite group \(G\), the group of units of the integral group ring \(\mathbb{Z} G\) contains the free product of two infinite cyclic groups unless \(G\) is Abelian or a Hamiltonian 2-group. \textit{Z. S. Marciniak} and \textit{S. K. Sehgal} [Proc. Am. Math. Soc. 125, No. 4, 1005-1009 (1997; Zbl 0869.16020)] proved that if the group of units contains a bicyclic unit \(u\) then the units \(u,u^*\) generate a subgroup isomorphic to the free product of two infinite cyclic groups. In this paper the authors prove that, for a finite group \(G\), the group of units of the integral group ring contains the free product of a cyclic group of prime order \(p\) and an infinite cyclic group if and only if the group \(G\) has a noncentral element of order \(p\). Moreover, this extends to torsion groups for the prime \(p=2\). The proof requires subtle group theoretical and module theoretical considerations.