Counting subgroups and topological group topologies

Abstract

Let G be an Abelian group with \G\ = a > , 6^{G) the set of subgroups of G, 88 the set of totally bounded topological group topologies on G,J?(y) the set of topological group topologies for which the character (= local weight) of (G, <^~) is equal to > , and &(y) = %CJ(y). We prove algebraic results and topological results, as follows. Algebra. Either |^(G)| = 2 a or |^(G)| = a. If \S?(G)\ = a then a = co. We describe and characterize those (countable) G such that \S?(G)\ = , and we give several examples. Topology. If < log() or > 2 , then &(y) = 0; otherwise 2 . If > 2 then Jt(y) = 0; if log() < < 2 then = 2 ; and if < < a then K()| = 2 .