On the index of a free abelian subgroup in the group of central units of an integral group ring

TL;DR

This paper establishes bounds on the index of a subgroup generated by a virtual basis within the central units of an integral group ring for certain finite groups, with applications to p-groups of small order.

Contribution

It provides new bounds on the index of subgroups generated by virtual bases in the central units of integral group rings for strongly monomial groups, including p-groups of order up to p^4.

Findings

Bound on the index of the subgroup in the central units

Determination of the rank of the central units group

Wedderburn decomposition of the rational group algebra

Abstract

Let $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ denote the group of central units in the integral group ring $\mathbb{Z}[G]$ of a finite group $G$. A bound on the index of the subgroup generated by a virtual basis in $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ is computed for a class of strongly monomial groups. The result is illustrated with application to the groups of order $p^{n}$, $p$ prime, $n \leq 4$. The rank of $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ and the Wedderburn decomposition of the rational group algebra of these $p$-groups have also been obtained.