Commuting probability for approximate subgroups of a finite group

TL;DR

This paper investigates how the structure of approximate subgroups in finite groups influences the probability that randomly chosen elements commute, providing insights into the algebraic properties of these subsets.

Contribution

It establishes new relationships between the structure of approximate subgroups and commuting probabilities in finite groups.

Findings

Approximate subgroups have specific commuting probability bounds.

Structural properties of A affect Pr(A,G) and Pr(A,A).

Results deepen understanding of subgroup behavior in finite groups.

Abstract

For subsets X,Y of a finite group G, we write Pr(X,Y) for the probability that two random elements x in X and y in Y commute. This paper addresses the relation between the structure of an approximate subgroup A of G and the probabilities Pr(A,G) and Pr(A,A).