4 problems

A group $H$ is called a UP-group if, for any two finite nonempty subsets $X,Y\subseteq H$, the product $XY$ contains an element that has a unique decomposition as the product of an…

Torsion-free group unique-product conjecture. Every torsion-free group is a UP-group.

Let $P$ be the Promislow group. For each positive integer $n$, let $D(n)$ denote the least word-radius of a non-unique-product $n$-element subset of $P$, when such subsets exist. P…

Unique product conjecture for uniform pro-$p$ groups. If $G$ is a uniform pro-$p$ group, then $G$ is a unique product group.