Group rings of finite strongly monomial groups: central units and primitive idempotents

TL;DR

This paper analyzes the structure of the group of central units and primitive idempotents in the integral group ring of finite strongly monomial groups, providing formulas, constructions, and generators for specific classes of such groups.

Contribution

It introduces formulas for the rank of central units, constructs virtual bases, and explicitly finds primitive idempotents for certain classes of finite strongly monomial groups.

Findings

Computed the rank of the central units group in $ ext{Z}G$.

Constructed a virtual basis for the central units group.

Explicitly constructed primitive idempotents for specific group classes.

Abstract

We compute the rank of the group of central units in the integral group ring $\Z G$ of a finite strongly monomial group $G$. The formula obtained is in terms of the strong Shoda pairs of $G$. Next we construct a virtual basis of the group of central units of $\Z G$ for a class of groups $G$ properly contained in the finite strongly monomial groups. Furthermore, for another class of groups $G$ inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of $\Q G$. Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of $\Z G$, this for metacyclic groups $G$ of the form $G=C_{q^m}\rtimes C_{p^n}$ with $p$ and $q$ different primes and the cyclic group $C_{p^n}$ of order $p^n$ acting faithfully on the cyclic group $C_{q^m}$ of order $q^m$.