On $2$-closed abelian permutation groups

TL;DR

This paper introduces a simple inductive criterion to determine when abelian permutation groups with cyclic transitive components are 2-closed, enhancing understanding of their structural properties.

Contribution

It provides a new, straightforward criterion for identifying 2-closed abelian permutation groups with cyclic transitive constituents.

Findings

Established an inductive criterion for 2-closedness.

Applied the criterion specifically to abelian groups with cyclic transitive parts.

Improved methods for analyzing the structure of 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.