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In this work we have considered the complexity of the different structures astopological group on Z. We collect some new results, as well as some knownresults 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-metrizabletopologies. We leave some open questions - for instance the problem of thecardinality of the families above.

Estructuras de grupo topol\'ogico en (Z,+)