On simultaneous conjugation of permutations

TL;DR

This paper proves a conjecture that if the commutator of two permutations in the symmetric group has many fixed points, then a permutation exists that inverts both, advancing understanding of permutation conjugation.

Contribution

It provides a proof for Neftin's conjecture relating fixed points of permutation commutators to simultaneous conjugation possibilities.

Findings

Confirmed the conjecture for permutations with commutators having at least n-4 fixed points.

Established conditions under which a permutation inverts both given permutations.

Enhanced understanding of the structure of symmetric groups and permutation conjugation.

Abstract

In this paper we give an affirmative answer to a conjecture proposed by Danny Neftin, that is, if the commutator of two permutations has at least n-4 fixed points where two permutations are in degree n symmetric group, then there exists a permutation in this symmetric inverting both of them.