21.34 (2026)

(Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A, B$ of $G$, there exists an element of $G$ which can be written uniquely as $ab$ with $a \in A$ and $b \in B$. A group $G$ is locally invariant orderable if $G$ admits a partial order $<$ such that for all $g, h \in G$ with $h \neq 1$, we have either $gh > g$ or $gh^{-1} > g$. Does there exist a unique product group which is not locally invariant orderable?