On quadratic rational Frobenius groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909572450484224 |
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| author | Pacifici, Emanuele Vergani, Marco |
| author_facet | Pacifici, Emanuele Vergani, Marco |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15841 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On quadratic rational Frobenius groups Pacifici, Emanuele Vergani, Marco Group Theory Representation Theory 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]. |
| title | On quadratic rational Frobenius groups |
| topic | Group Theory Representation Theory |
| url | https://arxiv.org/abs/2408.15841 |