On 2-closed abelian permutation groups

Abstract

A permutation group $G\le \text{Sym}(\Omega )$ is said to be 2-closed if no group H such that $G<H\le \text{Sym}(\Omega )$ 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.

Keywords:

• 2-Closed groups
• abelian groups
• permutation groups

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20B25
• 20B10

Acknowledgments

The authors thank S. Skresanov and A. Vasil’ev for their suggestions for improving the text, and are grateful to the anonymous referee for suggestions improving the presentation.

Notes

1 The cited statement was formulated for arbitrary intransitive groups with 2-closed transitive constituents.

Additional information

Funding

The research was supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613.