Enumeration of solvable cube-free groups and counting certain types of split extensions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911040684425216 |
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| author | Kumar, Prashun Venkataraman, Geetha |
| author_facet | Kumar, Prashun Venkataraman, Geetha |
| contents | A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this paper. Let $p$ be a prime and let $q = p^k$ for some positive integer $k$. We also give a formula for the number of conjugacy classes of the subgroups that are maximal amongst non-abelian solvable cube-free $p'$-subgroups of ${\rm GL}(2,q)$. Further, we find the exact number of split extensions of $P$ by $Q$ up to isomorphism of a given order where $P \in \{{\mathbb Z}_p \times {\mathbb Z}_p, {\mathbb Z}_{p^α}\}$, $p$ is a prime, $α$ is a positive integer and $Q$ is a cube-free abelian group of odd order such that $p \nmid |Q|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration of solvable cube-free groups and counting certain types of split extensions Kumar, Prashun Venkataraman, Geetha Group Theory 20E28, 20E34, 20E45, 20F99 A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this paper. Let $p$ be a prime and let $q = p^k$ for some positive integer $k$. We also give a formula for the number of conjugacy classes of the subgroups that are maximal amongst non-abelian solvable cube-free $p'$-subgroups of ${\rm GL}(2,q)$. Further, we find the exact number of split extensions of $P$ by $Q$ up to isomorphism of a given order where $P \in \{{\mathbb Z}_p \times {\mathbb Z}_p, {\mathbb Z}_{p^α}\}$, $p$ is a prime, $α$ is a positive integer and $Q$ is a cube-free abelian group of odd order such that $p \nmid |Q|$. |
| title | Enumeration of solvable cube-free groups and counting certain types of split extensions |
| topic | Group Theory 20E28, 20E34, 20E45, 20F99 |
| url | https://arxiv.org/abs/2504.12789 |