  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.
