The influence of the nilpotentlizers on group structur
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909119894519808 |
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| author | Ahmadkhah, N. Zarrin, M. |
| author_facet | Ahmadkhah, N. Zarrin, M. |
| contents | For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and the size of nilpotentizer of an element on finite group. In fact, we show that if there exists an element $x$ of $G$ such that $nil_G(x)$ generates a maximal subgroup of $G$ and the simple commutator of weight $2 ~~or ~~3$ of elements of $nil_G(x)$ is equal to $1$ or $|nil_G(x)|= p^n$, where $p$ is prime and $n=1, 2$. Then $G$ is a solvable group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The influence of the nilpotentlizers on group structur Ahmadkhah, N. Zarrin, M. Group Theory 20D60, 20D15, 20D10 For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and the size of nilpotentizer of an element on finite group. In fact, we show that if there exists an element $x$ of $G$ such that $nil_G(x)$ generates a maximal subgroup of $G$ and the simple commutator of weight $2 ~~or ~~3$ of elements of $nil_G(x)$ is equal to $1$ or $|nil_G(x)|= p^n$, where $p$ is prime and $n=1, 2$. Then $G$ is a solvable group. |
| title | The influence of the nilpotentlizers on group structur |
| topic | Group Theory 20D60, 20D15, 20D10 |
| url | https://arxiv.org/abs/2402.15916 |