Total closure for permutation actions of finite nonabelian simple groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909091028271104 |
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| author | Freedman, Saul D. Giudici, Michael Praeger, Cheryl |
| author_facet | Freedman, Saul D. Giudici, Michael Praeger, Cheryl |
| contents | For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $Ω$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ that leaves invariant each of the $G$-orbits in the induced action on $Ω\times\cdots\times Ω=Ω^k$. Each finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer $k$ such that $G$ is totally $k$-closed. We address the question of determining the closure number $k(G)$ for finite simple groups $G$. Prior to our work it was known that $k(G)=2$ for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $k(G)\geq3$ for all other finite simple groups. We determine the value for the alternating groups, namely $k(A_n)=n-1$. In addition, for all simple groups $G$, other than alternating groups and classical groups, we show that $k(G)\leq 7$. Finally, if $G$ is a finite simple classical group with natural module of dimension $n$, we show that $k(G)\leq n+2$ if $n \ge 14$, and $k(G) \le \lfloor n/3 + 12 \rfloor$ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on $n$ and the type of $G$) on the base sizes of the primitive actions of $G$, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_02347 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Total closure for permutation actions of finite nonabelian simple groups Freedman, Saul D. Giudici, Michael Praeger, Cheryl Group Theory 20B05 (Primary) 20E32, 20B15 (Secondary) For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $Ω$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ that leaves invariant each of the $G$-orbits in the induced action on $Ω\times\cdots\times Ω=Ω^k$. Each finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer $k$ such that $G$ is totally $k$-closed. We address the question of determining the closure number $k(G)$ for finite simple groups $G$. Prior to our work it was known that $k(G)=2$ for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $k(G)\geq3$ for all other finite simple groups. We determine the value for the alternating groups, namely $k(A_n)=n-1$. In addition, for all simple groups $G$, other than alternating groups and classical groups, we show that $k(G)\leq 7$. Finally, if $G$ is a finite simple classical group with natural module of dimension $n$, we show that $k(G)\leq n+2$ if $n \ge 14$, and $k(G) \le \lfloor n/3 + 12 \rfloor$ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on $n$ and the type of $G$) on the base sizes of the primitive actions of $G$, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups. |
| title | Total closure for permutation actions of finite nonabelian simple groups |
| topic | Group Theory 20B05 (Primary) 20E32, 20B15 (Secondary) |
| url | https://arxiv.org/abs/2206.02347 |