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Strange divisibility in groups and rings

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Abstract

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative ring, the number of Pythagorean triples (as well as four-tuples, etc.) of invertible elements is a multiple of the order of the multiplicative group.

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References

  1. K. S. Brown, The coset poset and probabilistic zeta function of a finite group, J. Algebra 225 (2000), 989–1012.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. J. Collins, Generating sequences of finite group, Senior Thesis, Cornell University Mathematics Department, 2010. http://www.math.cornell.edu/m/sites/default/files/imported/Research/SeniorTheses/2010/collinsThesis.pdf. (accessed January 1 2017).

  3. P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134–151.

    Article  MATH  Google Scholar 

  4. T. Hawkes, I. M. Isaacs, and M. Özaydin, On the Möbius function of a finite group, Rocky Mountain J. Math. 19 (1984), 1003–1034.

    Article  MATH  Google Scholar 

  5. C. Gordon and F. Rodriguez-Villegas, On the divisibility of #\({\rm {Hom}}(\Gamma , G)\) by \(|G|\), J. Algebra 350 (2012), 300–307.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. A. Klyachko and A. A. Mkrtchyan, How many tuples of group elements have a given property? With an appendix by Dmitrii V. Trushin, Int. J. Algebra Comp. 24 (2014), 413–428.

  7. C. Kratzer and J. Thévenaz, Fonction de Möbius d’un groupe fini et anneau de Burnside, Comment. Math. Helv. 59 (1984), 425–438.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Solomon, The solutions of equations in groups, Arch. Math. 20 (1969), 241–247.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. A. Wilson, The finite simple groups. graduate texts in mathematics, Springer, 251, London, 2009.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University (MSU), Leninskie Gory, Moscow, 119991, Russia

    Anton A. Klyachko

  2. School of Mathematics, University of Edinburgh, Room 5402, James Clerk Maxwell Building, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Anna A. Mkrtchyan

Authors
  1. Anton A. Klyachko
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  2. Anna A. Mkrtchyan
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Correspondence to Anton A. Klyachko.

Additional information

The work of A. A. Klyachko was supported by the Russian Foundation for Basic Research, Project No. 15-01-05823.

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Klyachko, A.A., Mkrtchyan, A.A. Strange divisibility in groups and rings. Arch. Math. 108, 441–451 (2017). https://doi.org/10.1007/s00013-016-1008-x

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  • DOI: https://doi.org/10.1007/s00013-016-1008-x

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