On quadrilaterals in the suborbital graphs of the normalizer

Document Type : Research Paper


1 Department of Mathematics, Faculty of Arts and Sciences, Niğde Ömer Halisdemir University, Niğde, Turkey

2 Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey

3 Department of Mathematics, Niğde Ömer Halisdemir University, Niğde, Turkey


n this paper‎, ‎we investigate suborbital graphs formed by $N\big(\Gamma_0(N)\big)$-invariant equivalence relation induced on $\hat{\mathbb{Q}}$‎. ‎Conditions for being an edge are obtained as a main tool‎, ‎then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated‎.


Main Subjects

[1] M. Akbaş and D. Singerman, The normalizer of Γ0 (N ) in P SL(2, R), Glasgow Math. J., 32 (1990) 317–327.
[2] M. Akbaş and D. Singerman, The signature of the normalizer of Γ0 (N ) in P SL(2, R), London Math. Soc., 165
(1992) 77–86.
[3] M. Akbaş and T. Başkan, Suborbital graphs for the normalizer of Γ0 (N ), Turkish J. Math., 20 (1996) 379–387.
[4] N. L. Biggs and A. T. White, Permutation groups and combinatorial structures, London Mathematical Society
Lecture Note Series, Cambridge University Press, Cambridge-New York, 33, 1979.
[5] K. S. Chua and M. L. Lang, Congruence subgroups associated to the monster, Experiment. Math., 13 (2004) 343–360.
[6] J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc., 11 (1979) 308-339.
[7] B. Ö. Güler, M. Beşenk, A. H. Değer and S. Kader, Elliptic elements and circuits in suborbital graphs, Hacet. J.
Math. Stat., 40 (2011) 203–210.
[8] B. Ö. Güler, T. Kör and Z. Şanlı, Solution to some congruence equations via suborbital graphs, Springerplus, 2016
(2016) 1–11.
[9] B. Ö. Güler, M. Beşenk and S. Kader, On congruence equations arising from suborbital graphs, Turkish. J. Math.,
43 (2019) 2396–2404.
[10] G. A. Jones, D. Singerman and K. Wicks, The modular group and generalized Farey graphs, London Math. Soc.
Lecture Note Ser., 160 (1991) 316–338.
[11] S. Kader, B. Ö. Güler and A. H. Değer, Suborbital graphs for a special subgroup of the normalizer, Iran. J. Sci.
Technol. A Sci., 34 (2010) 305–312.
[12] S. Kader, Circuits in Suborbital Graphs for The Normalizer, Graphs Combin., 33 (2017) 1531–1542.
[13] R. Keskin, Suborbital graphs for the normalizer of Γ0 (m), European J. Combin., 27 (2006) 193–206.
[14] R. Keskin and B. Demirtürk, On suborbital graphs for the normalizer of Γ0 (N ), Electron. J. Combin., 16 (2009)
pp. 18.
[15] J. Lehner and M. Newman, Weierstrass points of Γ0 (N ), Ann. of Math. (2), 79 (1964) 360–368.
[16] C. Maclachlan, Groups of units of zero ternary quadratic forms, Proc. Roy. Soc. Edinburgh Sect. A, 88 (1981)
[17] M. Magnus, Combinatorial group theory, Wiley, New York, 1966.
[18] H. E. Rose, A Course in Number Theory, Oxford University Press, 1982.
[19] C. C. Sims, Graphs and finite permutation groups, Math. Z., 95 (1967) 76-86.
[20] T. Tsuzuku, Finite Groups and Finite Geometries, Cambridge University Press, Cambridge, 1982.
Volume 9, Issue 3 - Serial Number 3
September 2020
Pages 147-159
  • Receive Date: 11 November 2019
  • Revise Date: 03 February 2020
  • Accept Date: 08 February 2020
  • Published Online: 01 September 2020