Kourovka Notebook Problem 21.34

Definitions and mathematical statement

A group $G$ has the unique product property if for every pair of nonempty finite sets $A,B\subseteq G$, some element $c\in AB$ has exactly one representation $c=ab$ with $(a,b)\in A\times B$. Equivalently, at least one fibre of the multiplication map $A\times B\to G$ is a singleton. The property does not require every product to have a unique representation.

A strict partial order $<$ on $G$ is an irreflexive transitive relation; comparability of every pair is not required. It is locally invariant in the sense used here if \[ \forall g\in G\quad\forall h\in G\setminus\{1\},\qquad g<gh\quad\text{or}\quad g<gh^{-1}. \] A group is locally invariant orderable if it admits at least one such partial order. The chosen alternative may depend on both $g$ and $h$. No global invariance under all left or right translations is demanded.

Does there exist a group with the unique product property which is not locally invariant orderable? Thus a positive answer requires proving the unique product condition for all finite $A,B$, while ruling out every strict partial order satisfying the displayed local alternatives.

The group is not assumed finite, finitely generated, or finitely presented. Its order relation is an auxiliary structure that may be chosen freely, not an order already supplied with the group. The question separates two existence properties; failing one particular attempted order does not prove failure of local invariant orderability.

Short English statement

Is there a unique product group admitting no partial order in which every nontrivial right translation moves a point upward in at least one direction?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2543 (KOU-21.34), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.34. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.