A note on the combinatorial derivation of nonsmall sets

Given an infinite group G and a subset A of G we let ∆(A) = {g ∈ G : |gA ∩ A| = ∞} (this is sometimes called the combinatorial derivation of A). A subset A of G is called: large if there exists a finite subset F of G such that FA = G; ∆-large if ∆(A) is large and small if for every large subset L of G, (G \ A) ∩ L is large. In this note we show that every nonsmall set is ∆-large, answering a question of Protasov.