Nondiscrete topological groups with many discrete subgroups

It is shown that for each positive integer n, there exists a nondiscrete Hausdorff topological group of cardinality ℵn with no proper subgroup of the same cardinality and with each proper subgroup discrete. This result is typical of those proved here using a method introduced by A.Yu. Olʹshanskii. It is also shown that there exists a continuum of pairwise algebraically nonisomorphic nondiscrete Hausdorff topological groups, each of which contains every finite group of odd order and has all of its proper subgroups finite.