Small subsets of groups

Given an infinite group G and an infinite cardinal κ ⩽ | G | , we say that a subset A of G is κ-large (κ-small) if there exists F ∈ [ G ] < κ such that G = F A ( G ∖ F A is κ-large for each F ∈ [ G ] < κ ). The subject of the paper is the family S κ of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by S κ . We study interrelations between κ-small and other ( P κ -small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each κ ⩾ ω . We show that [ G ] < κ is dual to S κ provided that either κ is regular and κ = | G | , or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group.