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Divisibility and laws in finite simple groups

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Abstract

We provide new bounds for the divisibility function of the free group \({\mathbf F}_2\) and construct short laws for the symmetric groups \({{\mathrm{Sym}}}(n)\). The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length of laws for finite simple groups of Lie type.

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Acknowledgments

This note was written during the trimester on Random Walks and Asymptotic Geometry of Groups at Institute Henri Poincaré in Paris. We are grateful to this institution for its hospitality. We are grateful to Mark Sapir for interesting remarks—especially about the comparison with the study of identities for associative algebras, and for bringing the work of Gimadeev–Vyalyi [10] to our attention. The second author thanks Emmanuel Breuillard and Martin Kassabov for valuable comments. The first author was supported by the Israel Science Foundation and the Jesselson Foundation. The second author was supported by ERC Starting Grant No. 277728.

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  1. Department of Mathematics, The Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    Gady Kozma

  2. Mathematisches Institut, U Leipzig, PF 100920, 04009, Leipzig, Germany

    Andreas Thom

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  1. Gady Kozma
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  2. Andreas Thom
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Kozma, G., Thom, A. Divisibility and laws in finite simple groups. Math. Ann. 364, 79–95 (2016). https://doi.org/10.1007/s00208-015-1201-4

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