Skip to main content
SpringerLink
Log in

Principal congruence subgroups of the Hecke groups and related results

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

In this paper, first, we determine the quotient groups of the Hecke groups H(λ q ), where q ≥ 7 is prime, by their principal congruence subgroups H p (λ q ) oflevel p, where p is also prime. We deal with the case of q = 7 separately, because of its close relation with the Hurwitz groups. Then, using the obtained results, we find the principal congruence subgroups of the extended Hecke groups \( \overline H \)(λ q ) for q ≥ 5 prime. Finally, we show that some of the quotient groups of the Hecke group H(λ q ) and the extended Hecke group \( \overline H \) (λ q ), q ≥ 5 prime, by their principal congruence subgroups H p (λ q ) are M*-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. O. Bizim and I.N. Cangül. Congruence subgroups of some Hecke groups. Bull. Inst. Math. Acad. Sinica, 30(2) (2002), 115–131.

    MATH  MathSciNet  Google Scholar 

  2. E. Bujalance, J.J. Etayo, J.M. Gamboa and G. Gromadzki. Automorphismsgroups of compact bordered Klein surfaces. A Combinatorial Approach. Lecture Notes in Math., 1439 (1990), Springer Verlag.

  3. E. Bujalance, F.J. Cirre and P. Turbek. GroupsactingonborderedKleinsurfaces with maximal symmetry. Proceedings of Groups St. Andrews 2001 in Oxford. Vol. I, 50–58, London Math. Soc. Lecture Note Ser., 304, Cambridge, U.K. Cambridge University Press (2003).

    Google Scholar 

  4. E. Bujalance, F.J. Cirre and P. Turbek. Subgroups of M*-groups. Q.J. Math., 54(1) (2003), 49–60.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Bujalance, F. J. Cirre and P. Turbek. Automorphism criteria for M*-groups. Proc. Edinb. Math. Soc., 47(2) (2004), 339–351.

    Article  MATH  MathSciNet  Google Scholar 

  6. I.N. Cangül. Normal subgroups of Hecke groups. Ph.D. Thesis, University of Southampton, Faculty of Mathematical Studies, December (1993).

  7. I.N. Cangül. The minimal polynomials of cos(2π/n) over ℚ. Problemy Mat., 15 (1997), 57–62.

    Google Scholar 

  8. I.N. Cangül and D. Singerman. Normal subgroups of Hecke groups and regular maps. Math. Proc. Camb. Phil. Soc., 123 (1998), 59–74.

    Article  MATH  Google Scholar 

  9. M.D.E. Conder. Hurwitz groups: a brief survey. Bull. Amer. Math. Soc., 23 (1990), 359–370.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Demirci and I.N. Cangül. A class of congruence subgroups of Hecke group H(λ5). Bull. Inst. Math. Acad. Sin. (N.S.), 1(4) (2006), 549–556.

    MATH  MathSciNet  Google Scholar 

  11. L.E. Dickson. Linear Groups with an Exposition of the Galois Field Theory, printed by Dover (1960).

  12. N. Greenleaf and C.L. May. Bordered Klein surfaces with maximal symmetry. Trans. Amer. Math. Soc., 274(1) (1982), 265–283.

    Article  MATH  MathSciNet  Google Scholar 

  13. E. Hecke. Über die bestimmung dirichletscher reihen durch ihre funktionalgleichungen. Math. Ann., 112 (1936), 664–699.

    Article  MathSciNet  Google Scholar 

  14. I. Ivrissimtzis and D. Singerman. Regular maps and principal congruence subgroups of Hecke groups. European J. Combin., 26(3–4) (2005), 437–456.

    Article  MATH  MathSciNet  Google Scholar 

  15. M.L. Lang. The signatures of the congruence subgroups G 0(τ) of the Hecke groups G 4 and G 6. Comm. Algebra, 28(8) (2000), 3691–3702.

    Article  MATH  MathSciNet  Google Scholar 

  16. M.L. Lang. The structure of the normalizers of the congruence subgroups of the Hecke Group G 5. Bull. London Math. Soc., 39(1) (2007), 53–62.

    Article  MATH  Google Scholar 

  17. M.L. Lang, C.H. Lim and S.P. Tan. Independent generators for congruence subgroups of Hecke groups. Math. Z., 220(4) (1995), 569–594.

    Article  MATH  MathSciNet  Google Scholar 

  18. M.L. Lang, C.H. Lim and S.P. Tan. Principal congruence subgroups of the Hecke groups. J. Number Theory, 85 (2000), 220–230.

    Article  MATH  MathSciNet  Google Scholar 

  19. A.M. Macbeath. Generators of the linear fractional groups. Proc. Symp. Pure Math., 12 A.M.S. (1969), 14–32.

    MathSciNet  Google Scholar 

  20. C.L. May. Automorphisms of compact Klein surfaces with boundary. Pacific J. Math., 59 (1975), 199–210.

    MATH  MathSciNet  Google Scholar 

  21. C.L. May. Large automorphism groups of compact Klein surfaces with boundary. Glasgow Math. J., 18 (1977), 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  22. C.L. May. A family of M*-groups. Canad. J. Math., 38(5) (1986), 1094–1109.

    MATH  MathSciNet  Google Scholar 

  23. C.L. May. Supersolvable M*-groups. Glasgow Math. J., 30(1) (1988), 31–40.

    Article  MATH  MathSciNet  Google Scholar 

  24. D.L. McQuillan. Classificationofnormalsubgroupsofthemodulargroup. Amer. J. Math., 87 (1965), 285–296.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Newman. Normal congruence subgroups of the modular group. Amer. J. Math., 85 (1963), 419–427.

    Article  Google Scholar 

  26. L.A. Parson. Normal Congruence subgroups of the Hecke groups G(2(1/2)) and G(3(1/2)). Pacific J. of Math., 70 (1977), 481–487.

    MATH  MathSciNet  Google Scholar 

  27. R. Sahin and O. Bizim. Some subgroups of the extended Hecke groups \( \overline H \)(λ q ). Acta Math. Sci. Ser. B, Engl. Ed., 23(4) (2003), 497–502.

    MATH  MathSciNet  Google Scholar 

  28. R. Sahin, O. Bizim and I.N. Cangül. Commutator subgroups of the extended Hecke groups \( \overline H \)(λ q ). Czechoslovak Math. J., 54(129), no. 1, (2004), 253–259.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Sahin, S. İkikardes and Ö. Koruoğlu. Some normal subgroups of the extended Hecke groups \( \overline H \)(λp). Rocky Mountain J. Math., 36(3) (2006), 1033–1048.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Sahin, S. İkikardes and Ö. Koruoğlu. Generalized M*-groups. Internat. J. Algebra Comput., 16(6) (2006), 1211–1219.

    Article  MATH  MathSciNet  Google Scholar 

  31. R. Sahin, S. İkikardes and Ö. Koruoğlu. Extended Hecke groups \( \overline H \) (λ q ) andtheir fundamental regions. Adv. Stud. Contemp. Math. (Kyungshang), 15(1) (2007), 87–94.

    MATH  MathSciNet  Google Scholar 

  32. D. Singerman. PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces. Proc. Edinb. Math. Soc., 30 (1987), 143–151.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fen-Edebiyat Fakultesi Matematik Bolumu, Balikesir Universitesi, 10145, Balikesir, Turkey

    Sebahattin Ikikardes & Recep Sahin

  2. Fen-Edebiyat Fakultesi Matematik Bolumu, Uludag Universitesi, 16059, Bursa, Turkey

    I. Naci Cangul

Authors
  1. Sebahattin Ikikardes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Recep Sahin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. Naci Cangul
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebahattin Ikikardes.

About this article

Cite this article

Ikikardes, S., Sahin, R. & Naci Cangul, I. Principal congruence subgroups of the Hecke groups and related results. Bull Braz Math Soc, New Series 40, 479–494 (2009). https://doi.org/10.1007/s00574-009-0023-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-009-0023-y

Keywords

Mathematical subject classification

Search

Navigation