Groups with unique product structures (Q1185961)

We say that a group \(G\) has a unique \(m\)-element product structure, where \(m\) is a positive integer, if there exists a subset \(S\) of \(G\) such that the product map \(f: S^ m\to G\), defined by \(f(x_ 1,x_ 2,\dots , x_ m)=x_ 1x_ 2\dots x_ m\), is a bijection. It is proved that a finite group with a unique \(m\)-element product structure for \(m\geq 2\) is trivial, i.e. has only one element. Based on the construction of free (3,2)-groups generated by the empty set the author gives examples of infinite groups with unique 2-element product structure.