Sylow subgroups for distinct primes and intersection of nilpotent subgroups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914288146317312 |
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| author | Lisi, Francesca Sabatini, Luca |
| author_facet | Lisi, Francesca Sabatini, Luca |
| contents | Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all $i$. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then we settle the conjecture in two opposite situations: symmetric and alternating groups of large degree and metanilpotent groups of odd order. Applications concerning the intersections of nilpotent subgroups are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sylow subgroups for distinct primes and intersection of nilpotent subgroups Lisi, Francesca Sabatini, Luca Group Theory 20D20, 20B30, 20Fxx Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all $i$. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then we settle the conjecture in two opposite situations: symmetric and alternating groups of large degree and metanilpotent groups of odd order. Applications concerning the intersections of nilpotent subgroups are discussed. |
| title | Sylow subgroups for distinct primes and intersection of nilpotent subgroups |
| topic | Group Theory 20D20, 20B30, 20Fxx |
| url | https://arxiv.org/abs/2505.21222 |