16.88 (2006)

(G. M. Bergman). The width of a group $G$ with respect to a generating set $X$ means the supremum over all $g \in G$ of the least length of a group word of $X$ expressing $g$. A group $G$ has finite width if the width of $G$ with respect to every generating set is finite. Does there exist a countably infinite group of finite width?

All known infinite groups of finite width (infinite permutation groups, infinite-dimensional general linear groups, and some other groups) are uncountable (G. M. Bergman, Bull. London Math. Soc., 38 (2006), 429–440, and references therein).