On 2-closed elusive permutation groups of degrees p2q and p2qr

Abstract

A transitive permutation group with no fixed point free elements of prime order is called elusive. A permutation group on a set Ω is said to be 2-closed if G is the largest subgroup of $\text{Sym}(\Omega )$ which leaves invariant each of the G-orbits for the induced action on $\Omega \times \Omega .$ There is a conjecture due to Marušič, Jordan, and Klin asserting that there is no elusive 2-closed permutation group. In this article, we give a proof of the conjecture for permutation groups of degrees $p^{2}q$ and $p^{2}qr,$ where p, q, and r are (not necessarily distinct) three primes.

Keywords:

• 2-closed permutation group
• elusive
• polycirculant conjecture

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20B05
• 20B10

Acknowledgments

The authors gratefully appreciate an anonymous referee for constructive comments and recommendations which definitely helped to improve the readability and quality of the article.

Additional information

Funding

The first author is financially supported by Iran National Science Foundation: INSF and University of Isfahan.