Involutions and free pairs of bicyclic units in integral group rings of non-nilpotent groups

Abstract:

If ${}^*\colon G\to G$ is an involution on the finite group $G$, then ${}^*$ extends to an involution on the integral group ring $\mathbb {Z}[G]$. In this paper, we consider whether bicyclic units $u\in \mathbb {Z}[G]$ exist with the property that the group $\langle u,u^*\rangle$, generated by $u$ and $u^*$, is free on the two generators. If this occurs, we say that $(u,u^*)$ is a free bicyclic pair. It turns out that the existence of $u$ depends strongly upon the structure of $G$ and on the nature of the involution. The main result here is that if $G$ is a non-nilpotent group, then for any involution, $\mathbb {Z}[G]$ contains a free bicyclic pair.