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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.10388 |
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| _version_ | 1866917722132054016 |
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| author | Hou, Dong-Dong Feng, Yan-Quan Leemans, Dimitri Qu, Hai-Peng |
| author_facet | Hou, Dong-Dong Feng, Yan-Quan Leemans, Dimitri Qu, Hai-Peng |
| contents | Let $(G,\{ρ_0, ρ_1, ρ_2\})$ be a string C-group of order $4p^m$ with type $\{k_1, k_2\}$ for $m \geq 2$, $k_1, k_2\geq 3$ and $p$ be an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$, $d(P)=2$, and up to duality, $p \mid k_1, 2p \mid k_2$. Moreover, we show that if $P$ is abelian, then $(G,\{ρ_0, ρ_1, ρ_2\})$ is tight and hence known. In the case where $P$ is nonabelian, we construct an infinite family of string C-group with type $\{p, 2p\}$ of order $4p^m$ where $m \geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | String C-groups of order $4p^m$ Hou, Dong-Dong Feng, Yan-Quan Leemans, Dimitri Qu, Hai-Peng Group Theory Combinatorics 20B25, 20D10, 52B10, 52B15 Let $(G,\{ρ_0, ρ_1, ρ_2\})$ be a string C-group of order $4p^m$ with type $\{k_1, k_2\}$ for $m \geq 2$, $k_1, k_2\geq 3$ and $p$ be an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$, $d(P)=2$, and up to duality, $p \mid k_1, 2p \mid k_2$. Moreover, we show that if $P$ is abelian, then $(G,\{ρ_0, ρ_1, ρ_2\})$ is tight and hence known. In the case where $P$ is nonabelian, we construct an infinite family of string C-group with type $\{p, 2p\}$ of order $4p^m$ where $m \geq 3$. |
| title | String C-groups of order $4p^m$ |
| topic | Group Theory Combinatorics 20B25, 20D10, 52B10, 52B15 |
| url | https://arxiv.org/abs/2407.10388 |