Local--global generation property of commutators in finite $π$-soluble groups
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916018884968448 |
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| author | Acciarri, Cristina Guralnick, Robert M. Khukhro, Evgeny Shumyatsky, Pavel |
| author_facet | Acciarri, Cristina Guralnick, Robert M. Khukhro, Evgeny Shumyatsky, Pavel |
| contents | For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $π$-group of automorphisms of a $π$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $π$-solubility. Earlier we obtained this type of results
for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local--global generation property of commutators in finite $π$-soluble groups Acciarri, Cristina Guralnick, Robert M. Khukhro, Evgeny Shumyatsky, Pavel Group Theory For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $π$-group of automorphisms of a $π$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $π$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups. |
| title | Local--global generation property of commutators in finite $π$-soluble groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2505.03017 |