Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi 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.

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We prove a removal lemma for systems of linear equations over finite fields: let X 1 , . . . , X m be subsets of the finite field F q and let A be a (k × m) matrix with coefficients in F q and rank k; if the linear system Ax = b has o(q m−k ) solutions with x i ∈ X i , then we can destroy all these solutions by deleting o(q) elements from each X i . This extends a result of Green [Geometric and Functional Analysis 15 (2) (2005), 340-376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.

We prove that there is ǫ > 0 and p0 > 0 such that for every prime p > p0, every subset S of Z/pZ which satisfies |2S| ≤ (2 + ǫ)|S| and 2(|2S|) − 2|S| + 3 ≤ p is contained in an arithmetic progression of length |2S| − |S| + 1. This is the first result of this nature which places no unnecessary restrictions on the size of S.

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the kdeterminantal of an integer (k × m) matrix A is coprime with the order n of a group G and the number of solutions of the system Ax = b with x1 ∈ X1, . . . , xm ∈ Xm is o(n m−k ), then we can eliminate o(n) elements in each set to remove all these solutions.algebraic removal lemma, hypergraph removal lemma, systems of linear equations.