Why are t.u.p groups torsion-free?

Question

I am new here so apologies if this is a dumb question. But I have recently encountered the idea of a two unique product group (t.u.p group). We take a group $G$ with two subsets $X, Y$. An element $g\in G$ can represented as a product in $X$ and $Y$ if $g=xy$ where $x\in X$ and $y\in Y$. It is represented uniquely if $g=x^\prime y^\prime$ $x^\prime\in X$, $y^\prime\in Y$ implies $x=x^\prime$, $y=y^\prime$. We suppose $|X|\geq 2$ and $|Y|\geq 2$ and then say $G$ is a t.u.p group if at least two elements can be represented uniquely in $X$ and $Y$. Now I have read in plenty of places that these groups must be torsion-free. Why is this? Perhaps I am not understanding the definition.