On the Removal Lemma for Linear Systems over Abelian Groups

Abstract: In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the number of solutions of the system $Ax=b$ with $x_1\in X_1,..., x_m\in X_m$ is $o(n^{m-k})$, then we can eliminate $o(n)$ elements in each set to remove all these solutions. This is a follow-up of our former paper 'A Removal Lemma for Systems of Linear Equations over Finite Fields' arXiv:0809.1846v1, which dealt with the case of finite fields.