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On a product of finite subsets in a torsion-free group. (English) Zbl 0697.20019
Let \(K\), \(M\) be finite subsets of a torsion-free group \(G\) such that \(| K|=k>1\), \(| M|=m>1\) and \(| KM|=k+m-1\). It is shown that there exist \(a,b,q\in G\) such that \(K=\{a,aq,...,aq^{k-1}\}\), \(M=\{b,qb,...,q^{m-1}b\}\).
Reviewer: V.Ufnarovskij

MSC:
20F05 Generators, relations, and presentations of groups
11B13 Additive bases, including sumsets
11P99 Additive number theory; partitions
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References:
[1] Freiman, G.A; Schein, B.M, Group and semigroup theoretic considerations inspired by inverse problems of the additive number theory, (), 121-140
[2] Kemperman, J.H.B, On complexes in a semigroup, Indag. math., 18, 247-254, (1956) · Zbl 0072.25605
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