Skip to main content
SpringerLink
Log in

On Suborbital Graphs for the Extended Modular Group \({\hat{\Gamma}}\)

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

In this paper, we show that the extended modular group \({\hat{\Gamma}}\) acts on \({\hat{\mathbb{Q}}}\) transitively and imprimitively. Then the number of orbits of \({\hat{\Gamma} _{0}(N)}\) on \({\hat{\mathbb{Q}}}\) is calculated and compared with the number of orbits of \({\Gamma _{0}(N)}\) on \({\hat{\mathbb{Q}}}\). Especially, we obtain the graphs \({\hat{G}_{u, N}}\) of \({\hat{\Gamma}_{0}(N)}\) on \({\hat{\mathbb{Q}}}\), for each \({N\in\mathbb{N}}\) and each unit \({u \in U_{N} }\), then we determine the suborbital graph \({\hat{F}_{u,N}}\). We also give the edge conditions in \({\hat{G}_{u, N}}\) and the necessary and sufficient conditions for a circuit to be triangle in \({\hat{F}_{u, N}.}\)

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. Akbaş M.: On suborbital graphs for the modular group. Bull. London Math. Soc. 33(6), 647–652 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akbaş, M.: The Normalizer of Modular subgroups. Ph.D. Thesis, University of Southampton, (1989)

  3. Akbaş M., Başkan T.: Suborbital Graphs for the Normalizer of \({\Gamma _{0}(N)}\). Turkish J. Math. 20(3), 379–387 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Biggs, N.L., White, A.T.: Permutation groups and combinatorial structures. London Math. Soc. Lecture Note Ser. 33, CUP, Cambridge (1979)

  5. Guler B.O., Besenk M., Deger A.H., Kader S.: Elliptic elements and circuits in suborbital graphs. Hacet. J. Math. Stat. 40(2), 203–210 (2011)

    MathSciNet  Google Scholar 

  6. Jones G.A., Singerman D., Wicks K.: The modular group and generalized farey graphs. London Math. Soc. Lecture Note Ser. 160, 316–338 (1991)

    MathSciNet  Google Scholar 

  7. Kader S., Guler B.O., Deger A.H.: Suborbital graphs for a special subgroup of the normalizer. IJST. Trans A. 34(A4), 305–312 (2010)

    MathSciNet  Google Scholar 

  8. Kulkarni R.S.: An arithmetic-geometric method in the study of the subgroups of the modular group. Am. J. Math. 113(6), 1053–1133 (1991)

    Article  MATH  Google Scholar 

  9. Neumann, P.M.: Finite permutation groups, edge-coloured graphs and matrices, topics. In: Curran, M.P.J. (ed.) Group Theory and Computation. Academic Press, London, New York, San Fransisco, (1977)

  10. Sims C.C.: Graphs and finite permutation groups. Math. Z. 95, 76–86 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tsuzuku T.: Finite Groups and Finite Geometries. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nigde University, Nigde, Turkey

    Serkan Kader

  2. Department of Mathematics, Rize University, Rize, Turkey

    Bahadır Özgür Güler

Authors
  1. Serkan Kader
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bahadır Özgür Güler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bahadır Özgür Güler.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kader, S., Güler, B.Ö. On Suborbital Graphs for the Extended Modular Group \({\hat{\Gamma}}\) . Graphs and Combinatorics 29, 1813–1825 (2013). https://doi.org/10.1007/s00373-012-1226-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-012-1226-3

Keywords

Search

Navigation