Skip to main content
SpringerLink
Log in

On some ranks of infinite groups

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

A group G has finite Hirsch-Zaicev rank r hz (G) = r if G has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors is exactly r. The authors discuss groups with finite Hirsch-Zaicev rank and the connection between this and groups having finite section p-rank for some prime p, or p=0. Groups all of whose abelian subgroups are of bounded rank are also discussed.

Keywords: p-rank, locally generalized radical group, Hirsch-Zaicev rank, torsion-free rank, rank

Mathematics Subject Classification (2000): 20F19, 20E25, 20E15

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. Baer, R., Heineken, H.: Radical groups of finite abelian subgroup rank. Illinois J. Math. 16, 533–580 (1972)

    Google Scholar 

  • 2. Čarin, V.S.: On groups of automorphisms of nilpotent groups. Ukrain. Mat. Ž. 6, 295–304 (1954)

    Google Scholar 

  • 3. Čarin, V.S.: Solvable groups of type A4. Mat. Sb. (N.S.) 52(94), 895–914 (1960)

  • 4. Dixon, M.R., Evans, M.J., Smith, H.: On groups with rank restrictions on subgroups. Groups St. Andrews 1997 in Bath, I, London Math. Soc. Lecture Note Ser. 260, Cambridge Univ. Press, Cambridge, 237–247 (1999)

  • 5. Dixon, M.R., Evans, M.J., Smith, H.: Embedding groups in locally (soluble-by-finite) simple groups. J. Group Theory 9, 383–395 (2006)

    Google Scholar 

  • 6. Franciosi, S., De Giovanni, F., Kurdachenko, L.A.: On groups with many almost normal subgroups. Ann. Mat. Pura Appl. 169, 35–65 (1995)

    Google Scholar 

  • 7. Hirsch, K.A.: On infinite soluble groups I. Proc. London Math. Soc. (2) 44, 53–60 (1938)

    Google Scholar 

  • 8. Kegel, O.H., Wehrfritz, B.A.F.: Locally Finite Groups. North-Holland Mathematical Library 3, North-Holland, Amsterdam, London (1973)

  • 9. Lucchini, A.: A bound on the number of generators of a finite group. Arch. Math. (Basel) 53, 4, 313–317 (1989)

    Google Scholar 

  • 10. Mal'cev, A.I.: On groups of finite rank. Mat. Sbornik 22, 351–352 (1948)

    Google Scholar 

  • 11. Mal'cev, A.I.: On certain classes of infinite soluble groups. Mat. Sbornik 28, 367–388 (1951) (Russian). English transl. Amer. Math. Soc. Translations.2, 1–21 (1956)

    Google Scholar 

  • 12. Merzljakov, Ju.I.: Locally solvable groups of finite rank. Algebra i Logika Sem. 3, 2, 5–16 (1964)

  • 13. Merzljakov, Ju.I.: Locally solvable groups of finite rank. II. Algebra i Logika 8, 686–690 (1969)

  • 14. Robinson, D.J.S.: Infinite soluble and nilpotent groups. Queen Mary College Mathematics Notes, Queen Mary College, London (1968)

  • 15. Robinson, D.J.S.: Finiteness Conditions and Generalized Soluble Groups vols. 1 and 2. Ergebnisse der Mathematik und ihrer Grenzgebiete 62 and 63, Springer-Verlag, Berlin, Heidelberg, New York, Band (1972)

  • 16. Robinson, D.J.S.: On the cohomology of soluble groups of finite rank. J. Pure and Applied Algebra 6, 155–164 (1975)

    Google Scholar 

  • 17. Robinson, D.J.S.: A new treatment of soluble groups with a finiteness condition on their abelian subgroups. Bull. London Math. Soc. 8, 113–129 (1976)

    Google Scholar 

  • 18. Robinson, D.J.S.: Soluble products of nilpotent groups. J. Algebra 98, 1, 183–196 (1986)

    Google Scholar 

  • 19. Sesekin, N.F.: On locally nilpotent groups without torsion. Mat. Sbornik N.S. 32(74), 407–442 (1953)

    Google Scholar 

  • 20. Šunkov, V.P.: On locally finite groups of finite rank. Algebra i Logika 10, 199–225 (1971) (Russian). English transl. in Algebra and Logic,10,127–142 (1971)

  • 21. Tomkinson, M.J.: FC-Groups. Pitman Publishing Limited, Boston, London, Melbourne (1984)

  • 22. Wehrfritz, B.A.F.: Infinite Linear Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 76, Springer-Verlag, New York, Heidelberg, Berlin (1973)

  • 23. Zaĭcev, D.I.: Solvable groups of finite rank. Groups with Restricted Subgroups (Russian). “Naukova Dumka”, Kiev, 115–130 (1971)

  • 24. Zaĭcev, D.I.: Solvable groups of finite rank. Algebra i Logika 16, 3, 300–312, 377 (1977)

  • 25. Zaĭcev, D.I.: Products of abelian groups. Algebra i Logika 19, (1980), 150–172, 250 (Russian). English transl. in Algebra and Logic 19, 94–106 (1980)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A.

    Martyn R. Dixon

  2. Department of Algebra, University of Dnepropetrovsk, Vulycya Naukova 13, Dnepropetrovsk 50, 49050, Ukraine

    Leonid A. Kurdachenko

  3. President, National Dnepropetrovsk University, Vulycya Naukova 13, Dnepropetrovsk 50, 49050, Ukraine

    Nikolay V. Polyakov

Authors
  1. Martyn R. Dixon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonid A. Kurdachenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nikolay V. Polyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dixon, M.R., Kurdachenko, L.A. & Polyakov, N.V. On some ranks of infinite groups. Ricerche mat. 56, 43–59 (2007). https://doi.org/10.1007/s11587-007-0004-7

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-007-0004-7

Keywords

Search

Navigation