A Note on Combinatorial Derivation

Given an infinite group $G$ and a subset $A$ of $G$ we let $\Delta(A) = {g \in G : |gA \cap A| =\infty}$ (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 $FA=G$. We show that given a large set $X$, and a decomposition $X=A_1\cup \mathrm{.}\mathrm{.}\mathrm{.}\cup A_n$, there must exist an $i$ such that $\mathrm{\Delta }(A_i)$ is large. This answers a question of Protasov. We also answer a number of related questions of Protasov.