On $2$-closed abelian permutation groups

Abstract

A permutation group $G\le\operatorname{Sym}(\Omega)$ is said to be $2$-closed if no group $H$ such that $G<H\le\operatorname{Sym}(\Omega)$ has the same orbits on $\Omega\times\Omega$ as $G$. A simple and efficient inductive criterion for the $2$-closedness is established for abelian permutation groups with cyclic transitive constituents.