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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.22410 |
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| _version_ | 1866917172134019072 |
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| author | Meng, Hangyang Yang, Yuting |
| author_facet | Meng, Hangyang Yang, Yuting |
| contents | Let $G$ be a finite group. For a prime $p$ and an integer $e \geq 0$, we denote by $Γ_{p,e}(G)$ the set of all pairs $(H, φ)$, where $H$ is a $p$-subgroup of $G$ of order greater than $p^e$ and $φ$ is a complex irreducible character of $H$. In this paper, we investigate the connected components of the poset $Γ_{p,e}(G)$. For the case $e = 0$, we prove that $Γ_{p,0}(G)$ is disconnected if and only if either $G$ has a strongly $p$-embedded subgroup, or every Sylow $p$-subgroup of $G$ contains a unique subgroup of order $p$. Furthermore, for $e = 1$ and $G$ a $p$-group, we show that the number of connected components of $Γ_{p,1}(G)$ equals the order of the intersection of all subgroups of $G$ of order $p^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connectivity of $p$-subgroup posets with irreducible characters Meng, Hangyang Yang, Yuting Group Theory Combinatorics 20C15, 20D15 Let $G$ be a finite group. For a prime $p$ and an integer $e \geq 0$, we denote by $Γ_{p,e}(G)$ the set of all pairs $(H, φ)$, where $H$ is a $p$-subgroup of $G$ of order greater than $p^e$ and $φ$ is a complex irreducible character of $H$. In this paper, we investigate the connected components of the poset $Γ_{p,e}(G)$. For the case $e = 0$, we prove that $Γ_{p,0}(G)$ is disconnected if and only if either $G$ has a strongly $p$-embedded subgroup, or every Sylow $p$-subgroup of $G$ contains a unique subgroup of order $p$. Furthermore, for $e = 1$ and $G$ a $p$-group, we show that the number of connected components of $Γ_{p,1}(G)$ equals the order of the intersection of all subgroups of $G$ of order $p^2$. |
| title | Connectivity of $p$-subgroup posets with irreducible characters |
| topic | Group Theory Combinatorics 20C15, 20D15 |
| url | https://arxiv.org/abs/2512.22410 |