19.26 (2018)

(Y. O. Hamidoune). Suppose that $A$ and $B$ are finite subsets of a group $G$ such that $|A| \geqslant 2$ and $|B| \geqslant 2$, and let $A \cdot_2 B$ denote the set of elements of $G$ which can be expressed in the form $ab$ for at least two different $(a, b) \in A \times B$. Is it true that $|A| + |B| - \frac{1}{2}|AB| - \frac{1}{2}|A \cdot_2 B| \leqslant \max\{2, |gH| \mid H \leqslant G, g \in G, gH \subseteq A \cdot_2 B\}$?