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 N-n, 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. Ol'shanskii. 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. (C) 1998 Published by Elsevier Science B.V.