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Automorphic Orbits

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Book cover Combinatorial Methods

Part of the book series: CMS Books in Mathematics ((CMSBM))

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Abstract

Automorphic orbits in a free group F n of finite rank are sets of the form \({\text{Or}}{{{\text{b}}}_{{{\text{Aut}}{{{\text{F}}}_{{\text{n}}}}}}}\)(u) = {υF n , υ = φ(u) for some φ ∈ Aut(F n ) and a fixed uF n }. One special and very interesting automorphic orbit is the set of all primitive elements (these are automorphic images of a free generator of F n ). In particular, the following problem, along with its generalizations [352], has been a source of inspiration for several people: Problem 4.0.1 ([352, 43]). If an endomorphism φ of a free group F n takes every primitive element to another primitive, is φ an automorphism?

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© 2004 Springer Science+Business Media New York

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Mikhalev, A.A., Shpilrain, V., Yu, JT. (2004). Automorphic Orbits. In: Combinatorial Methods. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21724-6_5

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  • DOI: https://doi.org/10.1007/978-0-387-21724-6_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2344-8

  • Online ISBN: 978-0-387-21724-6

  • eBook Packages: Springer Book Archive

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