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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Groups with many rewritable products


Authors: Mario Curzio, Patrizia Longobardi, Mercede Maj and Akbar Rhemtulla
Journal: Proc. Amer. Math. Soc. 115 (1992), 931-934
MSC: Primary 20F24
MathSciNet review: 1086580
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Abstract: For any integer $ n \geq 2$, denote by $ {R_n}$ the class of groups $ G$ in which every infinite subset $ X$ contains $ n$ elements $ {x_1}, \ldots ,{x_n}$ such that the product $ {x_1} \ldots {x_n} = {x_{\sigma (1)}} \cdots {x_{\sigma (n)}}$ for some permutation $ \sigma \ne 1$. The case $ n = 2$ was studied by B. H. Neumann who proved that $ {R_2}$ is precisely the class of centre-by-finite groups. Here we show that $ G \in {R_n}$ for some $ n$ if and only if $ G$ contains an FC-subgroup $ F$ of finite index such that the exponent of $ F/Z(F)$ is finite, where $ Z(F)$ denotes the centre of $ F$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1086580-X
PII: S 0002-9939(1992)1086580-X
Article copyright: © Copyright 1992 American Mathematical Society