On quadratic rational Frobenius groups

TL;DR

This work addresses the problem of classifying Frobenius groups that are quadratic rational, meaning all character value fields $\mathbb{Q}(\chi)$ have degree at most $2$ over $\mathbb{Q}$. The authors develop a framework using central units, Galois actions, and semi-inertia to relate quadratic rationality to uniform semi-rationality and to the structure of Frobenius kernels and complements. They provide a complete classification: solvable Frobenius complements must lie in a finite list (including $C_2$, $C_3$, $C_4$, $C_6$, $Q_8$, $H_1$, $H_2$, etc.), with a unique non-solvable case $H\cong SL_2(5)$, and show that all such groups are uniformly semi-rational. In the abelian-kernel case, they establish that semi-rationality, quadratic rationality, and uniform semi-rationality coincide, and they derive explicit kernel-exponent constraints; for odd-order complements the groups are inverse semi-rational. These results illuminate the Gruenberg-Kegel graphs realizable by quadratic rational groups and complete connections with prior work on cut and semi-rational groups.

Abstract

Let $G$ be a finite group and, for a given complex character $χ$ of $G$, let ${\mathbb{Q}}(χ)$ denote the field extension of ${\mathbb{Q}}$ obtained by adjoining all the values $χ(g)$, for $g\in G$. The group $G$ is called quadratic rational if, for every irreducible complex character $χ\in{\rm{Irr}}(G)$, the field ${\mathbb{Q}}(χ)$ 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].

Theorems & Definitions (72)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• proof
• Theorem 2.7
• Lemma 2.8