Abstract
Let δ be an automorphism of prime order p of the free group F n . Suppose δ has no fixed points and preserves the length of words. By σ := δ (m) we denote the automorphism of the free solvable group \({F_{n}/F_n^{(m)} }\) induced by δ. We show that every fixed point of σ has the form \({cc^{\sigma} \ldots c^{\sigma^{p-1}}}\), where \({c\in F_n^{(m-1)}/F_n^{(m)}}\). This is a generalization of some known results, including the Macedońska–Solitar Theorem [10].
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Acknowledgements
The author wishes to thank Vitaliy Sushchansky for his critical reading of this text and for many helpful remarks, due to which this paper is of a more general form than its first version. The author would also like to thank Czesław Bagiński who read the first version of this text and offered many valuable suggestions.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Tomaszewski, W. Fixed points of automorphisms preserving the length of words in free solvable groups. Arch. Math. 99, 425–432 (2012). https://doi.org/10.1007/s00013-012-0438-3
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DOI: https://doi.org/10.1007/s00013-012-0438-3