On the global breadth of finite groups with nontrivial partitions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866908419630301184 |
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| author | Muhie, Seid Kassaw Otera, Daniele Ettore Russo, Francesco G. |
| author_facet | Muhie, Seid Kassaw Otera, Daniele Ettore Russo, Francesco G. |
| contents | In a series of recent contributions on the notion of global breadth $\mathbf{B}(G)$ of a finite group $G$, it was interesting to observe the structural conditions arising from the classification of finite groups of $\mathbf{B}(G)=8$. This motivated the study of a new class of finite groups, namely $\mathcal{H}=\{G \ | \ G \ \mbox{satisfies the condition } \ |G| \le \mathbf{B}(G)(\mathbf{B}(G) + 1)\}$ and very little is known about $\mathcal{H}$. Here we focus on the groups with nontrivial partitions (according to the terminology of Baer, Kegel and Kontorovich), determining first that $\mathbf{B}(G)$ is achieved via the local breadth in connection with the order of maximal cyclic subgroups. Then we show that $\mathcal{H}$ contains projective special linear groups, projective general linear groups and Suzuki groups, supporting the conjecture that all finite groups with nontrivial partitions belong to $\mathcal{H}$. The presence of large families of simple groups in $\mathcal{H}$ is shown for the first time here. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19675 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the global breadth of finite groups with nontrivial partitions Muhie, Seid Kassaw Otera, Daniele Ettore Russo, Francesco G. Group Theory 20D15, 20D60, 20D15 In a series of recent contributions on the notion of global breadth $\mathbf{B}(G)$ of a finite group $G$, it was interesting to observe the structural conditions arising from the classification of finite groups of $\mathbf{B}(G)=8$. This motivated the study of a new class of finite groups, namely $\mathcal{H}=\{G \ | \ G \ \mbox{satisfies the condition } \ |G| \le \mathbf{B}(G)(\mathbf{B}(G) + 1)\}$ and very little is known about $\mathcal{H}$. Here we focus on the groups with nontrivial partitions (according to the terminology of Baer, Kegel and Kontorovich), determining first that $\mathbf{B}(G)$ is achieved via the local breadth in connection with the order of maximal cyclic subgroups. Then we show that $\mathcal{H}$ contains projective special linear groups, projective general linear groups and Suzuki groups, supporting the conjecture that all finite groups with nontrivial partitions belong to $\mathcal{H}$. The presence of large families of simple groups in $\mathcal{H}$ is shown for the first time here. |
| title | On the global breadth of finite groups with nontrivial partitions |
| topic | Group Theory 20D15, 20D60, 20D15 |
| url | https://arxiv.org/abs/2506.19675 |