Skip to main content
SpringerLink
Log in

On normal verbal embeddings of groups

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

Abstract

For the case of an arbitrary groupH and an arbitrary word setV, we establish a necessary and sufficient condition under which there exists a groupG such thatH is isomorphic to a normal subgroup\(\tilde H\) ofG such that\(\tilde H\) lies inV(G). This is a generalization of results of Burnside and Blackburn (concerning the cases of the commutator word and some much more special classes of groups) as well as of the first author (establishing a criterion for the case of one wordw and finitep-groupH). Some related special cases are considered.

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. Baer, “Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich,”Stud. Math.,5, 15–17 (1934).

    Google Scholar 

  2. N. Blackburn, “On prime power groups in which the derived group has two generators,”Proc. Cambridge Philos. Soc.,53, 19–27 (1957).

    MATH  MathSciNet  Google Scholar 

  3. W. Burnside, “On some properties of groups whose orders are powers of primes,”Proc. London Math. Soc. (2),11, 225–245 (1912).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Heineken, “Normal embeddings ofp-groups intop-groups,”Proc. Edinburgh Math. Soc.,35, 309–314 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Heineken and J. C. Lennox, “The subnormal embedding of complete groups,”J. Algebra,9, 435–445 (1984).

    Article  MathSciNet  Google Scholar 

  7. H. Heineken and V. H. Mikaelian, “Über dienV-Einbettbarkeit von den einigen Klassen der Gruppen,” in preparation.

  8. O. Hölder, “Bildung zusammengesetzter Gruppen,”Math. Ann.,46, 312–422 (1895).

    Article  Google Scholar 

  9. M. I. Kargapolov and Yu. I. Merzlyakov,Fundamentals of the Theory of Groups [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  10. V. H. Mikaelian, “Subnormal embedding theorems for groups,” to appear.

  11. D. W. Miller, “On a theorem of Hölder,”Amer. Math. Monthly,65, No. 4, 252–254 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  12. Hanna Neumann,Varieties of Groups, Springer Verlag, Berlin (1968).

    Google Scholar 

  13. D. J. S. Robinson,A Course in the Theory of Groups, Springer-Verlag, New York-Berlin-Heidelberg (1996).

    Google Scholar 

Download references

Authors
  1. H. Heineken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. H. Mikaelian
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 58, Algebra-12, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heineken, H., Mikaelian, V.H. On normal verbal embeddings of groups. J Math Sci 100, 1915–1924 (2000). https://doi.org/10.1007/BF02677503

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02677503

Keywords

Search

Navigation