关于至多具有五个非有理共轭类的群
On groups with at most five irrational conjugacy classes
AI总结：
本文研究至多含五个非有理共轭类的有限群，证明此时有理不可约特征标数与有理共轭类数相等，揭示了与少量有理不可约特征标群研究之间的对偶性，且结果不依赖有限单群分类定理。
摘要：
已有大量工作致力于研究具有少量有理共轭类或少量有理不可约特征标的群。本文考察相反的极端情形。设 $G$ 为有限群。给定 $G$ 的一个共轭类 $K$，若存在某个 $χ\in \operatorname{Irr}(G)$ 使得 $χ(K) \not \in \mathbb{Q}$，则称其为非有理的。我们的主要结果之一表明，当 $G$ 至多含有 $5$ 个非有理共轭类时，有 $|\operatorname{Irr}_{\mathbb{Q}}(G)| = |\operatorname{cl}_{\mathbb{Q}}(G)|$。这提示了与关于具有少量有理不可约特征标的群的已知结果和公开问题之间的某种对偶性。我们的结果不依赖于有限单群分类定理。
英文摘要：
Much work has been done to study groups with few rational conjugacy classes or few rational irreducible characters. In this paper we look at the opposite extreme. Let $G$ be a finite group. Given a conjugacy class $K$ of $G$, we say it is irrational if there is some $χ\in \operatorname{Irr}(G)$ such that $χ(K) \not \in \mathbb{Q}$. One of our main results shows that, when $G$ contains at most $5$ irrational conjugacy classes, then $|\operatorname{Irr}_{\mathbb{Q}}(G)| = |\operatorname{cl}_{\mathbb{Q}}(G)|$. This suggests some duality with the known results and open questions on groups with few rational irreducible characters. Our results are independent of the Classification of Finite Simple Groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
