A combinatorial proof of the Removal Lemma for Groups

TL;DR

This paper presents a new proof of the Removal Lemma for finite groups, extending its applicability beyond abelian groups and exploring potential extensions to systems of equations.

Contribution

It offers a combinatorial proof that generalizes the Removal Lemma from abelian to all finite groups, broadening its theoretical scope.

Findings

Extended the Removal Lemma to all finite groups

Provided a combinatorial proof alternative to Green's analytical approach

Discussed potential extensions to systems of equations

Abstract

Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemer\'edi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.