Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ1, τ2 on a group G are called transversal if the least upper bound τ1∨τ2 of τ1 and τ2 is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c2 pairwise transversal complete group topologies on G (so c2 maximal group topologies). Moreover, every abelian group G admits 2|G|2 pairwise transversal complete group topologies.