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Normal closure of one-variable equations in free groups


Author: C. Sibertin-Blanc
Journal: Proc. Amer. Math. Soc. 80 (1980), 34-38
MSC: Primary 20E05
DOI: https://doi.org/10.1090/S0002-9939-1980-0574504-2
MathSciNet review: 574504
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Abstract: Let $ w(x)$ be a one-variable equation in a free group F of finite rank. Lyndon has proved that it is possible to associate effectively to $ w(x)$ the set of its solutions, whereas Appel and Lorenc have provided a simpler representation of the set inferred. In this paper, we invert the problem and demonstrate that if the elements of any set $ S \subset F$ are solutions of an equation $ w(x)$, then $ w(x)$ belongs to the normal closure of finitely many short equations associated to S. A few consequences are given.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0574504-2
Keywords: One-variable equation, set of solutions, free generator, parametric word, normal closure, primitive element, recursively enumerable
Article copyright: © Copyright 1980 American Mathematical Society

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