Zelenyuk Y. Ultrafilters and topologies on groups

There is founded the theory of local left topological groups. It is proved that beta N has no nontrivial finite groups. Finite semigroups of ultrafilters of countable topological groups are described. It is constructed the partition of any Abelian group with a finite number of second order elements into two sets that don't contain cosets modulo infinite subgroups. It is proved that is impossible to construct the extremally disconnected topological group within ZFC. A classification of spaces of countable topological groups whose topology is determined by compact subsets is obtained. It is proved that all countable Abelian groups and fields contain infinite topologically free subsets.