Lutsenko I. Combinatorial sizes of subsets of groups

We classify the subsets of an infinite group by its combinatorial size and study an interplay between the subsets of distinct types. For every infinite group, it is constructed the 2-thin system of generators and a 4-thin subset $X$ such that $G=XX^{-1}$. We calculate a numbers of boolean group ideals and monogenerated boolean group ideals of countable groups. We prove that a lattice of group ideals on Abelian group is modular, and characterize the strongly prime ultrafilters with usage at sparse subsets. We describe the ideal generated by thin subsets of a countable group. We show that each vector ballean is uniquely determined by some vector ideal. It is characterized a closure by thin and by sparse subsets of every downward closed family of subsets. It is proved a completeness by thin subsets of the ideal of all small subsets of arbitrary infinite group.