On quadratic rational Frobenius groups

TL;DR

This paper classifies quadratic rational Frobenius groups, revealing their structure and properties, and connects these groups to broader classes like cut groups and uniformly semi-rational groups, advancing understanding of their algebraic and Galois-theoretic features.

Contribution

It provides a complete classification of Frobenius groups that are quadratic rational, linking their structure to properties of their integral group rings and Galois actions.

Findings

Every Frobenius quadratic rational group is uniformly semi-rational.

All generators of cyclic subgroups lie in at most two conjugacy classes.

The class coincides with the groups studied in previous work, completing their analysis.

Abstract

Let $G$ be a finite group and, for a given complex character $\chi$ of $G$, let ${\mathbb{Q}}(\chi)$ denote the field extension of ${\mathbb{Q}}$ obtained by adjoining all the values $\chi(g)$, for $g\in G$. The group $G$ is called quadratic rational if, for every irreducible complex character $\chi\in{\rm{Irr}}(G)$, the field ${\mathbb{Q}}(\chi)$ is an extension of ${\mathbb{Q}}$ of degree at most $2$. Quadratic rational groups have a nice characterization in terms of the structure of the group of central units in their integral group ring, and in fact they generalize the well-known concept of a cut group (i.e., a finite group whose integral group ring has a finite group of central units). In this paper we classify the Frobenius groups that are quadratic rational, a crucial step in the project of describing the Gruenberg-Kegel graphs associated to quadratic rational groups. It turns out that every Frobenius quadratic rational group is uniformly semi-rational, i.e., it satisfies the following property: all the generators of any cyclic subgroup of $G$ lie in at most two conjugacy classes of $G$, and these classes are permuted by the same element of the Galois group ${\rm{Gal}}({\mathbb{Q}}_{|G|}/{\mathbb{Q}})$ (in general, every cut group is uniformly semi-rational, and every uniformly semi-rational group is quadratic rational). We will also see that the class of groups here considered coincides with the one studied in [4], thus the main result of this paper also completes the analysis carried out in [4].