Modular permutations on \(\mathbb{Z}\). (Q957949)

Summary: The group \(\mathcal M\) of permutations \(\sigma\) of \(\mathbb{Z}\) for which an integer \(n=n(\sigma)>0\) exists such that \((z+n)\sigma=z\sigma+n\) for every \(z\in\mathbb{Z}\) is studied. \(\mathcal M\) is countably infinite locally (Abelian-by-finite) and contains all finitely generated (Abelian-by-finite) groups as subgroups. The commutator subgroup \(\mathcal M'\) is an infinite simple group and the quotient group \(\mathcal{M/M}'\) is isomorphic to \(\mathbb{Z}\). Finally, all Abelian groups that can be represented as modular permutation groups are determined: these are countable Abelian groups whose quotient over the torsion subgroup is free.