A combinatorial proof of the Removal Lemma for Groups

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.