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The group of all finite-state automorphisms of a regular rooted tree has a minimal generating set

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Abstract

We find some sufficient conditions under which the permutational wreath product of two groups has a minimal (irredundant) generating set. In particular we prove that for a regular rooted tree the group of all automorphisms and the group of all finite-state automorphisms of such a tree satisfy these conditions. Thereby we solve the problem that was stated by B. Csákány and F. Gécseg in 1965.

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Acknowledgments

The author gratefully acknowledges the many helpful suggestions of Ievgen Bondarenko, Volodymyr Nekrashevych, Andriy Oliynyk, and Wital Sushchansky during the preparation of the paper.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64, Volodymyrs’ka St., Kiev, 01601, Ukraine

    Yaroslav Lavrenyuk

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  1. Yaroslav Lavrenyuk
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Correspondence to Yaroslav Lavrenyuk.

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Lavrenyuk, Y. The group of all finite-state automorphisms of a regular rooted tree has a minimal generating set. Geom Dedicata 183, 59–67 (2016). https://doi.org/10.1007/s10711-016-0145-5

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  • DOI: https://doi.org/10.1007/s10711-016-0145-5

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