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

 

Primitive elements in free groups


Author: Martin J. Evans
Journal: Proc. Amer. Math. Soc. 106 (1989), 313-316
MSC: Primary 20F05
MathSciNet review: 952315
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Abstract: Let $ {F_n}$ denote the free group of rank $ n$ generated by $ {x_1},{x_2}, \ldots ,{x_n}$. We say that $ y \in {F_n}$ is a primitive element of $ {F_n}$ if it is contained in a set of free generators of $ {F_n}$. In this note we construct, for each integer $ n \geq 4$, an $ (n - 1)$-generator group $ H$ that has an $ n$-generator, $ 2$-relator presentation $ H = \langle {x_1}, \ldots ,{x_n}\vert{r_1},{r_2}\rangle $ such that the normal closure of $ \{ {r_1},{r_2}\} $ in $ {F_n}$ does not contain a primitive element of $ {F_n}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0952315-1
PII: S 0002-9939(1989)0952315-1
Article copyright: © Copyright 1989 American Mathematical Society