8.9 (1982)

(C. Chou). We say that a group $G$ has property $P$ if for every finite subset $F$ of $G$, there is a finite subset $S \supset F$ and a subset $X \subset G$ such that $x_1S \cap x_2S$ is empty for any $x_1, x_2 \in X$, $x_1 \neq x_2$, and $G = \bigcup_{x \in X} Sx$. Does every group have property $P$?