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=FA (G∖FA is κ-large for each F∈[G]<κ). The subject of the paper is the family Sκ of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by Sκ. We study interrelations between κ-small and other (Pκ-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 Sκ 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.