Central units of integral group rings of monomial groups

TL;DR

This paper investigates the structure of central units in integral group rings of certain monomial groups, proving that Bass units generate a subgroup of finite index and determining the rank of the central units for generalized strongly monomial groups.

Contribution

It establishes that Bass units generate a subgroup of finite index in the central units for subgroup closed monomial groups and determines the rank of the central units for generalized strongly monomial groups.

Findings

Bass units generate a subgroup of finite index in the central units.

The rank of the central units is explicitly determined.

The results are expressed in terms of generalized strong Shoda pairs.

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 $G$.