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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Commutators as products of squares


Authors: Roger C. Lyndon and Morris Newman
Journal: Proc. Amer. Math. Soc. 39 (1973), 267-272
MSC: Primary 20F05
DOI: https://doi.org/10.1090/S0002-9939-1973-0314997-5
MathSciNet review: 0314997
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Abstract: It is shown that if $ G$ is the free group of rank 2 freely generated by $ x$ and $ y$, then $ xy{x^{ - 1}}{y^{ - 1}}$ is never the product of two squares in $ G$, although it is always the product of three squares in $ G$. It is also shown that if $ G$ is the free group of rank $ n$ freely generated by $ {x_1},{x_2}, \cdots ,{x_n}$, then $ x_1^2x_2^2 \cdots x_n^2$ is never the product of fewer than $ n$ squares in $ G$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0314997-5
Article copyright: © Copyright 1973 American Mathematical Society