Length parameters of finite groups and their Hall subgroups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917475338158080 |
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| author | Khukhro, Evgeny Shumyatsky, Pavel |
| author_facet | Khukhro, Evgeny Shumyatsky, Pavel |
| contents | Let $π$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $π$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The proof uses the fact, obtained in [4] using the classification of finite simple groups, that a finite simple group of order divisible by $p$ cannot have a nilpotent Hall $\{2,p\}$-subgroup. As a corollary, it is proved that if in addition $H$ is soluble, then the non-$p$-soluble length of $G$ is bounded above by $2l_2(H)+1$, where $l_2(H)$ is the $2$-length of $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08596 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Length parameters of finite groups and their Hall subgroups Khukhro, Evgeny Shumyatsky, Pavel Group Theory Let $π$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $π$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The proof uses the fact, obtained in [4] using the classification of finite simple groups, that a finite simple group of order divisible by $p$ cannot have a nilpotent Hall $\{2,p\}$-subgroup. As a corollary, it is proved that if in addition $H$ is soluble, then the non-$p$-soluble length of $G$ is bounded above by $2l_2(H)+1$, where $l_2(H)$ is the $2$-length of $H$. |
| title | Length parameters of finite groups and their Hall subgroups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.08596 |