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On Generating Sets of Infinite Symmetric Group

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It is shown that in the group of all bijections of an infinite set, certain families of subsets related to the cardinality of some proper subsets are generating. A criterion for generating an infinite symmetric group by sets of this kind is obtained. As a special case of these results, the number of generating sets of the symmetric group S(Ω) is described for Ω = ℤ.

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References

  1. G. M. Bergman, “Generating infinite symmetric groups,” Bull. London Math. Soc., 38, 429–440 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  2. J. D. Dixon and B. M. Mortimer, Permutation Groups, Graduate Texts in Mathematics, Springer (1988).

    Google Scholar 

  3. F. Galvin, “Generating countable sets of permutations,” J. London Math. Soc. (2), 51, 230–242 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Macpherson and P. M. Neumann, “Subgroups of infinite symmetric groups,” J. London Math. Soc. (2), 64–84 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Moran, “Conjugacy classes whose square is an infinite symmetric group,” Trans. Amer. Math. Soc., 316, 493–522 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Droste and W. Ch. Holland, “Generating automorphism groups of chains,” Forum Math., 17, No. 4, 699–710 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Droste, “Classes of universal words for the infinite symmetric groups,” Algebra Universalis, 20, 205–216 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Droste and R. Gobel, “Uncountable cofinalities of permutation groups,” J. London Math. Soc. (2) 71, 335–344 (2005).

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Chebyshev Laboratory, St.Petersburg State University, St.Petersburg, Russia

    A. V. Semenov

  2. Laboratory for Continuous Mathematical Education, St.Petersburg, Russia

    A. D. Denisova

Authors
  1. A. V. Semenov
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  2. A. D. Denisova
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Correspondence to A. V. Semenov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 500, 2021, pp. 177–187.

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Semenov, A.V., Denisova, A.D. On Generating Sets of Infinite Symmetric Group. J Math Sci 272, 464–469 (2023). https://doi.org/10.1007/s10958-023-06436-8

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  • DOI: https://doi.org/10.1007/s10958-023-06436-8

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