Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits 2 c pairwise transversal complete group topologies on G (so 2 c maximal group topologies). Moreover, every abelian group G admits 2 2 | G | pairwise transversal complete group topologies.