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Primitive and measure-preserving systems of elements on the varieties of metabelian and metabelian profinite groups

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Abstract

Given a free metabelian group S of finite rank r, r ≥ 2, we prove that a system of elements g 1, ..., g n ∈ S for n = 1 or n = r preserves measure on the variety of all metabelian groups if and only if the system is primitive. Similar results hold for a free profinite group \(\hat S\) and the variety of finite metabelian groups for each n, 1 ≤ n ≤ r. Some corollaries to these theorems are derived.

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References

  1. Puder D., “On primitive words. II: measure preservation,” arXiv:1104.3991v2[math.GR] 21 Apr., 2011, pp. 1–31.

  2. Puder D. and Pazanchevski O., “Measure preserving words are primitive,” arXiv:1202.32691v1 [math.GR] 15 Feb., 2012, pp. 1–39.

  3. Gupta Ch. K. and Timoshenko E. I., “Generating elements for groups of the form \(\mathfrak{A}_m \mathfrak{A}_n\): the criterion and induction,” Algebra and Logic, 40, No. 3, 137–143 (2001).

    Article  MathSciNet  Google Scholar 

  4. Timoshenko E. I., Endomorphisms and Universal Theories of Soluble Groups [in Russian], Izdat. NGTU, Novosibirsk (2011).

    Google Scholar 

  5. Gupta C. K. and Roman’kov V. A., “Finite separability of tameness and primitivity in certain relatively free groups,” Comm. Algebra, 23, No. 11, 4101–4108 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  6. Wilson J., Profinite Groups, Clarendon Press, Oxford (1998).

    MATH  Google Scholar 

  7. Skornyakov L. A. (Ed.), General Algebra. Vol. 1 [in Russian], Nauka, Moscow (1990).

    Google Scholar 

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Authors and Affiliations

  1. Novosibirsk State Technical University, Novosibirsk, Russia

    E. I. Timoshenko

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  1. E. I. Timoshenko
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Correspondence to E. I. Timoshenko.

Additional information

To the 70th anniversary of Victor Danilovich Mazurov.

Original Russian Text Copyright © 2013 Timoshenko E.I.

The author was supported by the Russian Foundation for Basic Research (Grant 12-01-00084) and the Ministry for Education and Science of the Russian Federation (Grant 14.B37.21.0359).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 1, pp. 199–207, January–February, 2013.

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Timoshenko, E.I. Primitive and measure-preserving systems of elements on the varieties of metabelian and metabelian profinite groups. Sib Math J 54, 152–158 (2013). https://doi.org/10.1134/S0037446613010199

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  • DOI: https://doi.org/10.1134/S0037446613010199

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