On a generalisation of Cameron's base size conjecture
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866911391254839296 |
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| author | Anagnostopoulou-Merkouri, Marina |
| author_facet | Anagnostopoulou-Merkouri, Marina |
| contents | Let $G\leqslant {\rm Sym}(Ω)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $Ω$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of $G$, denoted by $b(G, Ω)$. Equivalently, $b(G, Ω)$ is the minimal positive integer $k$ such that $G$ has a regular orbit on the Cartesian product $Ω^k$. A well-known conjecture of Cameron from the 1990s asserts that if $G$ is an almost simple primitive group and $H$ is a so-called non-standard subgroup, then $b(G, Ω) \leqslant 7$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action of degree $24$. This conjecture was settled in a series of papers by Burness et al. (2007-11).
In this paper, we complete the proof of a natural generalisation of Cameron's conjecture. Our main result states that if $G$ is an almost simple group and $H_1, \ldots, H_k$ are any non-standard maximal subgroups of $G$ with $k \geqslant 7$, then $G$ has a regular orbit on $G/H_1 \times \cdots \times G/H_k$, noting that Cameron's original conjecture corresponds to the special case where the $H_i$ are pairwise conjugate subgroups. In addition, we show that the same conclusion holds with $k = 6$, unless $G = {\rm M}_{24}$ and each $H_i$ is isomorphic to ${\rm M}_{23}$. For example, this means that if $G$ is a simple exceptional group of Lie type and $H_1, \ldots, H_6$ are proper subgroups of $G$, then there exist elements $g_i \in G$ such that $\bigcap_i H_i^{g_i} = 1$. By applying recent work in a joint paper with Burness, we may assume $G$ is a group of Lie type and our proof uses probabilistic methods based on fixed point ratio estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a generalisation of Cameron's base size conjecture Anagnostopoulou-Merkouri, Marina Group Theory 20B05 Let $G\leqslant {\rm Sym}(Ω)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $Ω$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of $G$, denoted by $b(G, Ω)$. Equivalently, $b(G, Ω)$ is the minimal positive integer $k$ such that $G$ has a regular orbit on the Cartesian product $Ω^k$. A well-known conjecture of Cameron from the 1990s asserts that if $G$ is an almost simple primitive group and $H$ is a so-called non-standard subgroup, then $b(G, Ω) \leqslant 7$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action of degree $24$. This conjecture was settled in a series of papers by Burness et al. (2007-11). In this paper, we complete the proof of a natural generalisation of Cameron's conjecture. Our main result states that if $G$ is an almost simple group and $H_1, \ldots, H_k$ are any non-standard maximal subgroups of $G$ with $k \geqslant 7$, then $G$ has a regular orbit on $G/H_1 \times \cdots \times G/H_k$, noting that Cameron's original conjecture corresponds to the special case where the $H_i$ are pairwise conjugate subgroups. In addition, we show that the same conclusion holds with $k = 6$, unless $G = {\rm M}_{24}$ and each $H_i$ is isomorphic to ${\rm M}_{23}$. For example, this means that if $G$ is a simple exceptional group of Lie type and $H_1, \ldots, H_6$ are proper subgroups of $G$, then there exist elements $g_i \in G$ such that $\bigcap_i H_i^{g_i} = 1$. By applying recent work in a joint paper with Burness, we may assume $G$ is a group of Lie type and our proof uses probabilistic methods based on fixed point ratio estimates. |
| title | On a generalisation of Cameron's base size conjecture |
| topic | Group Theory 20B05 |
| url | https://arxiv.org/abs/2511.08705 |