On the removal lemma for linear systems over Abelian groups

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the kk-determinantal of an integer (k×m)(k×m) matrix AA is coprime with the order nn of a group GG and the number of solutions of the system Ax=bAx=b with x1∈X1,…,xm∈Xmx1∈X1,…,xm∈Xm is o(nm−k)o(nm−k), then we can eliminate o(n)o(n) elements in each set to remove all these solutions.