Does there exist a torsion-free group $(G,.)$ with the following property?

There exists finite, non-empty subsets $S_1,S_2\subset G$ such that for all $s \in S_1.S_2$ there are at least two solutions to the equation $s=s_1.s_2$ where $s_1\in S_1$ and $s_2\in S_2$.


I imagine that such a group $(G,.)$ does exist. If you've found one, please could you explain how you constructed it also?


The motivation for this question is to prove that the group ring (for example over $\mathbb{Z}$) of any torsion-free group has no zero divisors, as conjectured by Kaplansky. If such $S_1$ and $S_2$ never exist then Kaplansky's conjecture would be true for lots of rings (any ring with no zero divisors in fact).

Thank you in advance.


These are the "unique product groups". Not every torsion-free group satisfies this property. See Promislow, S. David, A simple example of a torsion-free, nonunique product group. Bull. London Math. Soc. 20 (1988), no. 4, 302–304.

  • $\begingroup$ Thanks a lot! And googling finds lots more papers with many more examples and references. Exactly what I wanted. $\endgroup$ – user30022 Mar 2 '13 at 0:00
  • $\begingroup$ In fact Promislow's example is somewhat better than others since in his example $S_1=S_2$. $\endgroup$ – user6976 Mar 2 '13 at 0:06

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