Skip to main content
SpringerLink
Log in

A problem from the kourovka notebook on embeddings of the group \( \mathbb{Q} \)

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Answering the question of de la Harpe and Bridson in the Kourovka Notebook, Problem 14.10(b), we construct explicit embeddings of the additive group of rational numbers \( \mathbb{Q} \) in a finitely generated group G. In fact, the group G is 2-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Dark, “On subnormal embedding theorems for groups,” J. London Math. Soc., 43, 387–390 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Hall, “The Frattini subgroups of finitely generated groups,” Proc. London Math. Soc. (3), 11, 327–352 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Hall, “On the embedding of a group in a join of given groups,” J. Aust. Math. Soc., 17, 434–495 (1974).

    Article  MATH  Google Scholar 

  4. G. Higman, B. H. Neumann, and H. Neumann, “Embedding theorems for groups,” J. London Math. Soc., 24, 247–254 (1949).

    Article  MathSciNet  Google Scholar 

  5. The Kourovka Notebook. Unsolved Problems in Group Theory, 14th edition (V. D. Mazurov and E. I. Khukhro, Eds.), Ross. Akad. Nauk, Sib. Otd. Inst. Mat. Novosibirsk (1999).

  6. A. G. Kurosh, The Theory of Groups, Chelsea Publ., New York (1960).

    Google Scholar 

  7. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin–New York (1977).

    MATH  Google Scholar 

  8. V. H. Mikaelian, “Subnormal embedding theorems for groups,” J. London Math. Soc. (2), 62, No. 2, 398–406 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. H. Mikaelian and H. Heineken, “On normal verbal embeddings of groups,” J. Math. Sci., 100, No. 1, 1915–1924 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. H. Mikaelian, “On varieties of groups generated by wreath products of abelian groups,” in: Abelian Groups, Rings, and Modules (Perth, 2000), Contemp. Math., 273, Amer. Math. Soc., Providence, Rhode Island (2001), pp. 223–238.

  11. V. H. Mikaelian, “On embeddings of countable generalized soluble groups in two-generated groups,” J. Algebra, 250, 1–17 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. V. H. Mikaelian, “Two problems on varieties of groups generated by wreath products,” Int. J. Math. Math. Sci., 31, No. 2, 65–75 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. H. Mikaelian, “An embedding construction for ordered groups,” J. Aust. Math. Soc., 74, No. 3, 379–392 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. H. Mikaelian, “On wreath products of finitely generated abelian groups,” in: Advances in Group Theory 2002, Aracne, Rome (2003), pp. 13–24.

  15. V. H. Mikaelian, “On normal verbal embeddings of some classes of groups,” Beitrage Algebra Geom., 45, No. 2, 501–516 (2004).

    MathSciNet  MATH  Google Scholar 

  16. V. H. Mikaelian, “Infinitely many not locally soluble SI*-groups,” Ric. Mat., 52, No. 1, 1–19 (2003).

    MathSciNet  MATH  Google Scholar 

  17. V. H. Mikaelian, “On embedding properties of SD-groups,” Int. J. Math. Math. Sci., Nos. 1–4, 65–76 (2004).

  18. V. H. Mikaelian, “On a problem on explicit embeddings of the group \( \mathbb{Q} \),” Int. J. Math. Math. Sci., No. 13, 2119–2123 (2005).

    Article  MathSciNet  Google Scholar 

  19. V. H. Mikaelian, “Metabelian varieties of groups and wreath products of abelian groups,” J. Algebra, 313, No. 2, 455–485 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. H. Mikaelian, “On finitely generated soluble non-Hopfian groups, an application to a problem of Neumann,” Int. J. Algebra Comput., 17, Nos. 5-6, 1107–1113 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  21. V. H. Mikaelian, “SD-groups and embeddings,” Arm. J. Math., 1, No. 3, 23–42 (2008).

    MathSciNet  Google Scholar 

  22. V. H. Mikaelian, “On finitely generated soluble non-Hopfian groups, Fundam. Prikl. Mat., 14, No. 8, 185–202 (2008).

    Google Scholar 

  23. V. H. Mikaelian, “Normal embeddings and wreath products of groups,” Fundam. Prikl. Mat. (in preparation).

  24. V. H. Mikaelian, “A geometrical interpretation of infinite wreath powers,” (in preparation).

  25. B. H. Neumann, “On ordered groups,” Am. J. Math., 71, 1–18 (1949).

    Article  MATH  Google Scholar 

  26. B. H. Neumann and H. Neumann, “Embedding theorems for groups,” J. London Math. Soc., 34, 465–479 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  27. P. M. Neumann, “On the structure of standard wreath products of groups,” Math. Z., 84, 343–373 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Yu. Ol’shanskiĭ, Geometry of Defining Relations in Groups, Math. Appl., 70, Kluwer Academic Publishers, Dordrecht (1991).

    Book  Google Scholar 

  29. D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts Math., 80, Springer-Verlag, New York (1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yerevan State University, Yerevan, Armenia

    V. H. Mikaelian

Authors
  1. V. H. Mikaelian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. H. Mikaelian.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 74, Proceedings of the International Conference “Modern Algebra and Its Applications” (Batumi, 2010), Part 1, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mikaelian, V.H. A problem from the kourovka notebook on embeddings of the group \( \mathbb{Q} \) . J Math Sci 186, 781–784 (2012). https://doi.org/10.1007/s10958-012-1033-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-1033-7

Keywords

Search

Navigation