Central units of integral group rings of monomial groups

Abstract

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units $\mathcal{Z}(\mathcal{U}(\mathbb{Z}G))$ of the integral group ring $\mathbb{Z}G$ for a subgroup closed monomial group $G$ with the property that every cyclic subgroup of order not a divisor of $4$ or $6$ is subnormal in $G$. If $G$ is a generalized strongly monomial group, then it is shown that the group generated by generalized Bass units contains a subgroup of finite index in $\mathcal{Z}(\mathcal{U}(\mathbb{Z}G))$. Furthermore, for a generalized strongly monomial group $G$, the rank of $\mathcal{Z}(\mathcal{U}(\mathbb{Z}G))$ is determined. The formula so obtained is in terms of generalized strong Shoda pairs of $G$.