On groups with square-free gcd of character degree and codegree
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913940737359872 |
|---|---|
| author | Aldahleh, Karam Kappler, Alan Makur, Neil Yang, Yong |
| author_facet | Aldahleh, Karam Kappler, Alan Makur, Neil Yang, Yong |
| contents | Let $G$ be a finite group and $χ$ be an irreducible character of $G$. The codegree of $χ$ is defined as $χ^c(1) =\frac{|G: \kerχ|}{χ(1)}$. In a paper by Gao, Wang, and Chen, it was shown that $G$ cannot satisfy the condition that $\gcd(χ(1),χ^c(1))$ is prime for all $χ\in\text{Irr}(G)^\#$. We generalize this theorem by solving one of Guohua Qian's unsolved problems on character codegrees. Qian inquires about the structure of non-solvable finite groups with square-free $\gcd$ instead. We prove that if $G$ is such that $\gcd(χ(1),χ^c(1))$ is square-free for every irreducible character $χ$, then $G/\text{Sol}(G)$ is isomorphic to one among a particular list of almost simple groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On groups with square-free gcd of character degree and codegree Aldahleh, Karam Kappler, Alan Makur, Neil Yang, Yong Group Theory 20C15 Let $G$ be a finite group and $χ$ be an irreducible character of $G$. The codegree of $χ$ is defined as $χ^c(1) =\frac{|G: \kerχ|}{χ(1)}$. In a paper by Gao, Wang, and Chen, it was shown that $G$ cannot satisfy the condition that $\gcd(χ(1),χ^c(1))$ is prime for all $χ\in\text{Irr}(G)^\#$. We generalize this theorem by solving one of Guohua Qian's unsolved problems on character codegrees. Qian inquires about the structure of non-solvable finite groups with square-free $\gcd$ instead. We prove that if $G$ is such that $\gcd(χ(1),χ^c(1))$ is square-free for every irreducible character $χ$, then $G/\text{Sol}(G)$ is isomorphic to one among a particular list of almost simple groups. |
| title | On groups with square-free gcd of character degree and codegree |
| topic | Group Theory 20C15 |
| url | https://arxiv.org/abs/2507.03100 |