Skip to main content
SpringerLink
Log in

The class of groups all of whose subgroups with lesser number of generators are free is generic

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

It is shown that, in a certain statistical sense, in almost every group withm generators andn relations (withm andn chosen), any subgroup generated by less thanm elements (which need not belong to the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked groups.

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. V. S. Guba, “Conditions under which 2-generated groups in small cancellation groups are free,”Izv. Vyssh. Uchebn. Zaved. Mat., [Soviet Math. J. (Iz. VUZ)], No. 7, 12–19 (1986).

    MATH  MathSciNet  Google Scholar 

  2. Kourov Notebook. (Unsolved Problems of Group Theory) [in Russian], 11th ed., Nauka, Novosibirsk (1990).

  3. S. W. Margolis and J. C. Meakin, “Free inverse monoids and graph immersions,”Int. J. Algebra and Comput. 3, No. 1, 79–100 (1993).

    MathSciNet  Google Scholar 

  4. Roger C. Lyndon and Paul E. Shupp,Combinatorial Group Theory, Springer-Verlag, Berlin-Heidelberg-New York (1977).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, USSR

    G. N. Arzhantseva & A. Yu. Ol'shanskii

Authors
  1. G. N. Arzhantseva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Ol'shanskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 489–496, April, 1996.

The research of the second author was partially supported by the Russian Foundation for Basic Research under grant No. 94-0101541 and by the International Science Foundation under grant No. MID000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arzhantseva, G.N., Ol'shanskii, A.Y. The class of groups all of whose subgroups with lesser number of generators are free is generic. Math Notes 59, 350–355 (1996). https://doi.org/10.1007/BF02308683

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02308683

Keywords

Search

Navigation