Some remarks on the orbit dimension of transitive groups and on the metric dimension of Johnson graphs
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| Format: | Preprint |
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2026
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| _version_ | 1866914475671552000 |
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| author | Drera, Alice Spiga, Pablo |
| author_facet | Drera, Alice Spiga, Pablo |
| contents | The orbit dimension $σ(G)$ (also called the separation number or rigidity index) of a permutation group $G$ with domain $Ω$ is the minimum cardinality of a subset $S \subseteq Ω$ such that, for any two distinct elements $ω,ω'\in Ω$, there exists $α\in S$ for which $ω$ and $ω'$ lie in distinct orbits of the stabilizer $G_α$.
In this paper, we first observe that if $G$ is transitive, then $σ(G)\le |Ω|-r+1$, where $r$ is the rank of $G$, and we obtain strong structural information on the groups for which equality holds.
Next, we investigate the orbit dimension in the case where $G$ is the symmetric group of degree $n$, acting on the set of $k$-subsets of $\{1,\ldots,n\}$. In this case, this invariant equals the metric dimension of Johnson graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13887 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some remarks on the orbit dimension of transitive groups and on the metric dimension of Johnson graphs Drera, Alice Spiga, Pablo Combinatorics Group Theory The orbit dimension $σ(G)$ (also called the separation number or rigidity index) of a permutation group $G$ with domain $Ω$ is the minimum cardinality of a subset $S \subseteq Ω$ such that, for any two distinct elements $ω,ω'\in Ω$, there exists $α\in S$ for which $ω$ and $ω'$ lie in distinct orbits of the stabilizer $G_α$. In this paper, we first observe that if $G$ is transitive, then $σ(G)\le |Ω|-r+1$, where $r$ is the rank of $G$, and we obtain strong structural information on the groups for which equality holds. Next, we investigate the orbit dimension in the case where $G$ is the symmetric group of degree $n$, acting on the set of $k$-subsets of $\{1,\ldots,n\}$. In this case, this invariant equals the metric dimension of Johnson graphs. |
| title | Some remarks on the orbit dimension of transitive groups and on the metric dimension of Johnson graphs |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2604.13887 |