On Topologies on the Group $({\mathbb{Z}}_p{)}^{\mathbb{N}}$

It is proved that, on any Abelian group of infinite cardinality $\mathbf{m}$, there exist precisely $2^{2^{\mathbf{m}}}$ nonequivalent bounded Hausdorff group topologies. Under the continuum hypothesis, the number of nonequivalent compact and locally compact Hausdorff group topologies on the group $({\mathbb{Z}}_p{)}^{\mathbb{N}}$ is determined.