On quadratic rational Frobenius groups

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].