In this work we have considered the complexity of the different structures as topological group on Z. We collect some new results, as well as some known results on the group of the integers in order to present: -A family of $2^\cont$ precompact Hausdorff. -A wide family of metrizable non-precompact, non-complete group topologies. -A wide family of complete non-metrizable topologies. We leave some open questions - for instance the problem of the cardinality of the families above.

Comment: in Spanish. Bolet\'in de la Sociedad de Profesores Puig Adam. Volumen 95, 2013