On Elusive Permutation Groups of Square-Free Degree

Abstract

An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order. In this article we prove that each minimal normal subgroup of an elusive permutation group with square-free degree is abelian. As an almost immediate consequence, it follows that there are no elusive permutation groups of degree 2pq where p, q are distinct odd primes.

Key Words:

• Elusive permutation groups
• Fixed point free elements of prime order
• Polycirculant conjecture

2000 Mathematics Subject Classification:

• 20B05

ACKNOWLEDGMENT

The author is grateful to Michael Giudici for helpful discussions on this topic.

Notes

Communicated by M. R. Dixon.