On the minimal degree and base size of finite primitive groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913572614832128 |
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| author | Mastrogiacomo, Fabio |
| author_facet | Mastrogiacomo, Fabio |
| contents | Let $G$ be a finite permutation group acting on $Ω$. A base for $G$ is a subset $B \subseteq Ω$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $μ(G)$, is the smallest cardinality of the support of a non trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $μ(G)b(G)\leq n \log n$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the minimal degree and base size of finite primitive groups Mastrogiacomo, Fabio Group Theory Let $G$ be a finite permutation group acting on $Ω$. A base for $G$ is a subset $B \subseteq Ω$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $μ(G)$, is the smallest cardinality of the support of a non trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $μ(G)b(G)\leq n \log n$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant. |
| title | On the minimal degree and base size of finite primitive groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2411.03469 |