21.58 (2026)

We say that a product $XY = \{xy \mid x \in X, y \in Y\}$ of two subsets $X, Y$ of a group $G$ is direct if for every $z \in XY$ there are unique $x \in X, y \in Y$ such that $z = xy$. Is there an infinite group $G$ such that every subset $A \subseteq G$ satisfies the following property: all the maximal subsets $B$ for which the product $AB$ is direct have the same cardinality?

Note that for checking the property for a given infinite group $G$, it suffices to consider only those subsets $A \subseteq G$ for which $|A| = |G \setminus A|$. Indeed, the property is equivalent to $A^{-1}A \cap B B^{-1} = \{1\}$ and $A^{-1}AB = G$, and these imply $|G| = |A||B|$, since $G$ is infinite. Now, if $|A| < |G \setminus A|$, then $|A| < |G|$, and so $|B| = |G|$; and if $|A| > |G \setminus A|$, then $A^{-1}A = G$, and so $|B| = 1$, for all $B$ satisfying the property.