关于二次有理 Frobenius 群
On quadratic rational Frobenius groups
AI总结：
本文分类了二次有理 Frobenius 群，证明其均为一致半有理群，并表明该类群与文献 [4] 研究的群类一致，从而完成了对二次有理群 Gruenberg-Kegel 图描述的关键一步。
摘要：
设 $G$ 为有限群，对于 $G$ 的给定复特征标 $χ$，令 ${\mathbb{Q}}(χ)$ 表示通过添加所有值 $χ(g)$（$g\in G$）而得到的 ${\mathbb{Q}}$ 的域扩张。如果对于每个不可约复特征标 $χ\in{\rm{Irr}}(G)$，域 ${\mathbb{Q}}(χ)$ 都是 ${\mathbb{Q}}$ 的次数至多为 $2$ 的扩张，则称群 $G$ 为二次有理群。二次有理群在其整群环的中心单位群的结构方面具有一个良好的刻画，并且事实上它们推广了众所周知的 cut 群概念（即整群环的中心单位群为有限群的有限群）。在本文中，我们对二次有理的 Frobenius 群进行分类，这是描述与二次有理群相关的 Gruenberg-Kegel 图的项目中的关键一步。结果表明，每个 Frobenius 二次有理群都是一致半有理的，即它满足以下性质：$G$ 的任意循环子群的所有生成元都位于 $G$ 的至多两个共轭类中，并且这些类被 Galois 群 ${\rm{Gal}}({\mathbb{Q}}_{|G|}/{\mathbb{Q}})$ 的同一个元素置换（一般地，每个 cut 群都是一致半有理的，并且每个一致半有理群都是二次有理的）。我们还将看到，这里考虑的群类与文献 [4] 中研究的群类一致，因此本文的主要结果也完成了文献 [4] 中所进行的分析。
英文摘要：
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].
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