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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 51 Issue: 3

Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$

Nazlı Yazıcı Gözütok(KARADENIZ TECHNICAL UNIVERSITY)0000-0002-3645-0623
Bahadır Özgür Güler(KARADENIZ TECHNICAL UNIVERSITY)0000-0003-3131-3643
Pages
—
Publication Date
2022-06-01
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In order to obtain graphs, we first define an imprimitive action of $\Gamma _B(N)$ on $\widehat{\mathbb{Q}}$ using the group $H_B(N)$ and then we obtain some properties of the graphs arising from this action.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we investigate suborbital graphs $G_{u,n}$ of the normalizer $\Gamma_B(N)$ of $\Gamma_0(N)$ in $PSL(2,\mathbb{R})$ for $N= 2^\alpha 3^\beta$, where $\alpha=0,2,4,6$ and $\beta =1,3$.
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Key Findings & Data:In each of these cases, the normalizer becomes a triangle group and the graph arising from the action of the normalizer contains hexagonal circuits.
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Conclusion & Impact:In order to obtain graphs, we first define an imprimitive action of $\Gamma _B(N)$ on $\widehat{\mathbb{Q}}$ using the group $H_B(N)$ and then we obtain some properties of the graphs arising from this action.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:normalizer · suborbital graph · hexagon

Abstract

In this paper, we investigate suborbital graphs $G_{u,n}$ of the normalizer $\Gamma_B(N)$ of $\Gamma_0(N)$ in $PSL(2,\mathbb{R})$ for $N= 2^\alpha 3^\beta$, where $\alpha=0,2,4,6$ and $\beta =1,3$. In each of these cases, the normalizer becomes a triangle group and the graph arising from the action of the normalizer contains hexagonal circuits. In order to obtain graphs, we first define an imprimitive action of $\Gamma _B(N)$ on $\widehat{\mathbb{Q}}$ using the group $H_B(N)$ and then we obtain some properties of the graphs arising from this action.

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Cite This Article

Gözütok, N. Y., Güler, B. Ö. (2022). Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$. Hacettepe Journal of Mathematics and Statistics, Volume: 51 Issue: 3. https://doi.org/10.15672/hujms.824436

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References (21)

  1. M. Akbaş and D. Singerman, The signature of the normalizer of $\Gamma _{0}(N)$ in $PSL(2,R)$,London Math. Soc. 165, 77–86, 1992.
  2. N.L. Biggs and A.T. White, Permutation groups and combinatorial structures, London Math. Soc. Lec. Not. Ser., 33rd ed. CUP, Cambridge, 1979.
  3. I.N. Cangül and D. Singerman, Normal subgroups of Hecke groups and regular maps, Math. Proc. Camb. Phil. Soc. 123, 59–74, 1998.
  4. K.S. Chua and M.L Lang, Congruence subgroups associated to the monster, Experiment. Math. 13 (3), 343–360, 2004.
  5. J.H. Conway and S.P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11, 308–339, 1977.
  6. H.S.M. Coxeter and W.O.F. Moser, Generators and Relations for Discrete Groups, 4th ed. Springer-Verlag, 1984.
  7. H.M. Farkas and I. Kra, Theta constants, Riemann surfaces and the modular group, Graduate Texts in Mathematics, 37, AMS, 2001.
  8. B.Ö. Güler, M. Beşenk and S. Kader, On congruence equations arising from suborbital graphs, Turk. J. Math. 43 (5), 2396–2404, 2019.
  9. I. Ivrissimtzis, D. Singerman and J. Strudwick, From farey fractions to the Klein quartic and beyond, Ars Math. Comtemp. 20 (1), 37–50, 2021.
  10. G.A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (3), 273–307, 1978.
  11. I. Ivrissimtzis and D. Singerman, Regular maps and principal congruence subgroups of Hecke groups, Eur. J. Comb. 26, 437–456, 2005.
  12. G.A. Jones and D. Singerman, Complex Functions, an Algebraic and Geometric Viewpoint, CUP, 1987.
  13. S. Kader, Circuits in suborbital graphs for the normalizer, Graphs Combin. 33 (6), 1531–1542, 2017.
  14. C. Maclachlan, Groups of units of zero ternary quadratic forms, Proc. Royal Soc. 88(A), 141-157, Edinburgh, 1981.
  15. C.C. Sims, Graphs and finite permutation groups, Math. Z. 95, 76-86, 1967.
  16. D. Singerman, Universal tessellations, Rev. Mat. Univ. Complut. 1, 111–123, 1988.
  17. D. Singerman and J. Strudwick, Petrie polygons, Fibonacci sequences and Farey maps, Ars Math. Contemp. 10 (2), 349–357, 2016.
  18. D. Singerman and J. Strudwick, The Farey maps modulo n, Acta Math. Univ. Comen. 89 (1), 39–52, 2020.
  19. J. Siran, How symmetric can maps on surfaces be?, Surveys in Combinatorics, 161– 238, London Math. Soc. Lec. Not. Ser. 409, CUP, Cambridge, 2013.
  20. N. Yazıcı Gözütok, U. Gözütok and B.Ö. Güler, Maps corresponding to the subgroups $\Gamma_0(N)$ of the modular group, Graphs Combin. 35 (6), 1695–1705, 2019.
  21. N. Yazıcı Gözütok and B.O. Güler, Quadrilateral cell graphs of the normalizer with signature $(2,4,\infty)$, Stud. Sci. Math. Hung. 57 (3), 408–425, 2020.