Supersolvable subgroups of order divisible by 3
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913809945329664 |
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| author | Beltrán, Antonio Shao, Changguo |
| author_facet | Beltrán, Antonio Shao, Changguo |
| contents | We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying the above condition on maximal subgroups, then either $G$ is a $3'$-group or $G/{\bf O}_{3'}(G)$ is isomorphic to ${\rm PSL}_2(2^p)$ for an odd prime $p$, where ${\bf O}_{3'}(G)$ denotes the largest normal $3'$-subgroup of $G$. Furthermore, in the latter case, ${\bf O}_{3'}(G)$ is nilpotent and ${\bf O}_2(G)\leq {\bf Z}(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Supersolvable subgroups of order divisible by 3 Beltrán, Antonio Shao, Changguo Group Theory 20D05, 20E28 We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying the above condition on maximal subgroups, then either $G$ is a $3'$-group or $G/{\bf O}_{3'}(G)$ is isomorphic to ${\rm PSL}_2(2^p)$ for an odd prime $p$, where ${\bf O}_{3'}(G)$ denotes the largest normal $3'$-subgroup of $G$. Furthermore, in the latter case, ${\bf O}_{3'}(G)$ is nilpotent and ${\bf O}_2(G)\leq {\bf Z}(G)$. |
| title | Supersolvable subgroups of order divisible by 3 |
| topic | Group Theory 20D05, 20E28 |
| url | https://arxiv.org/abs/2504.18289 |