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Proceedings of the American Mathematical Society

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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}$.

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

  • [1] M. J. Dunwoody, Relation modules, Bull. London Math. Soc. 4 (1972), 151-155. MR 0327915 (48:6257)
  • [2] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0577064 (58:28182)
  • [3] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Interscience, New York, 1966.
  • [4] M. S. Montgomery, Left and right inverses in group algebras, Bull. Amer. Math. Soc. 75 (1969), 539-540. MR 0238967 (39:327)
  • [5] G. A. Noskov, Primitive elements in a free group, Mat. Zametki 30(4) (1981), 497-500. MR 638422 (83e:20039)

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Article copyright: © Copyright 1989 American Mathematical Society

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