On the Codegree graphs of finite groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915559783792640 |
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| author | Chen, Jiyong Du, Ni Li, Leyi |
| author_facet | Chen, Jiyong Du, Ni Li, Leyi |
| contents | The codegree of an irreducible character $χ$ of a finite group $G$ is defined as $|G:\kerχ|/χ(1)$. The codegree graph $Γ(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, where two distinct primes $p$ and $q$ are adjacent if and only if $pq$ divides the codegree of some irreducible character of $G$. In this paper, we prove that a graph can occur as a codegree graph $Γ(G)$ of some finite group $G$ if and only if its complement is triangle-free and $3$-colorable. This generalizes the known characterization for codegree graphs from solvable groups to all finite groups. As an application, we give a full classification of all groups for which $Γ(G)$ is a $5$-cycle. We also investigate conditions under which the codegree graph coincides with or differs from the prime graph for solvable groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Codegree graphs of finite groups Chen, Jiyong Du, Ni Li, Leyi Group Theory 20C15 The codegree of an irreducible character $χ$ of a finite group $G$ is defined as $|G:\kerχ|/χ(1)$. The codegree graph $Γ(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, where two distinct primes $p$ and $q$ are adjacent if and only if $pq$ divides the codegree of some irreducible character of $G$. In this paper, we prove that a graph can occur as a codegree graph $Γ(G)$ of some finite group $G$ if and only if its complement is triangle-free and $3$-colorable. This generalizes the known characterization for codegree graphs from solvable groups to all finite groups. As an application, we give a full classification of all groups for which $Γ(G)$ is a $5$-cycle. We also investigate conditions under which the codegree graph coincides with or differs from the prime graph for solvable groups. |
| title | On the Codegree graphs of finite groups |
| topic | Group Theory 20C15 |
| url | https://arxiv.org/abs/2510.15791 |