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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 $s_2\in S_2$.

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

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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.

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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.

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  • $\begingroup$ 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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