On finite $d$-maximal groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915140351295488 |
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| author | Lucchini, Andrea Sabatini, Luca Stanojkovski, Mima |
| author_facet | Lucchini, Andrea Sabatini, Luca Stanojkovski, Mima |
| contents | Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up to isomorphism and only partial results have been proven for larger $d$. In this work, we prove that a $d$-maximal group is supersolvable and we give a characterization of $d$-maximality in terms of so-called maximal $(p,q)$-pairs. Moreover, we classify the maximal $(p,q)$-pairs of small rank obtaining, as a consequence, a full classification of the isomorphism classes of $3$-maximal finite groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_16254 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On finite $d$-maximal groups Lucchini, Andrea Sabatini, Luca Stanojkovski, Mima Group Theory Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up to isomorphism and only partial results have been proven for larger $d$. In this work, we prove that a $d$-maximal group is supersolvable and we give a characterization of $d$-maximality in terms of so-called maximal $(p,q)$-pairs. Moreover, we classify the maximal $(p,q)$-pairs of small rank obtaining, as a consequence, a full classification of the isomorphism classes of $3$-maximal finite groups. |
| title | On finite $d$-maximal groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2305.16254 |