Some remarks on the orbit dimension of transitive groups and on the metric dimension of Johnson graphs

TL;DR

This paper studies the orbit dimension of transitive permutation groups, establishing bounds and structural properties, and explores its relation to the metric dimension of Johnson graphs, especially for symmetric groups acting on k-subsets.

Contribution

It provides bounds on the orbit dimension for transitive groups, characterizes cases of equality, and links this invariant to Johnson graph metric dimension for symmetric groups.

Findings

For transitive groups, (G)  |\u03a9|-r+1, where r is the rank.

Structural insights into groups where equality holds.

Orbit dimension equals Johnson graph metric dimension for symmetric groups.

Abstract

The orbit dimension $\sigma(G)$ (also called the separation number or rigidity index) of a permutation group $G$ with domain $\Omega$ is the minimum cardinality of a subset $S \subseteq \Omega$ such that, for any two distinct elements $\omega,\omega'\in \Omega$, there exists $\alpha\in S$ for which $\omega$ and $\omega'$ lie in distinct orbits of the stabilizer $G_\alpha$. In this paper, we first observe that if $G$ is transitive, then $\sigma(G)\le |\Omega|-r+1$, 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.