Article

# On the cardinality of sumsets in torsion-free groups

09/2010; DOI: 10.1112/blms/bds032

Source: arXiv

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**ABSTRACT:**Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.03/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.01/2013;

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