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) = {υ ∈ F n , υ = φ(u) for some φ ∈ Aut(F n ) and a fixed u ∈ F 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
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