Almost commuting permutations are near commuting permutations

TL;DR

This paper proves that permutations which nearly commute are close to truly commuting permutations under the Hamming distance, extending the result to multiple permutations and specific subclasses like even permutations.

Contribution

It establishes the stability of the commutator in permutations, showing that almost commuting permutations are near actual commuting permutations, including for k-tuples and restricted classes.

Findings

Almost commuting permutations are near commuting permutations.

The stability result extends to k-tuples of permutations.

Applicable to subclasses such as even permutations.

Abstract

We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations. Our result extends to $k$-tuples of almost commuting permutations, for any given $k$, and allows restrictions, for instance, to even permutations.