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Journals / Hacettepe Journal of Mathematics and Statistics / 2011 / Cilt: 40 - Sayı: 1

THE IMAGES UNDER THE MODULAR GROUP AND EXTENDED MODULAR GROUP

Özden KORUOĞLU
Pages
15–20
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

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

Using these images, unlike known methods, we can show that the extended modular group takes the upper half of the complex plane into H.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper we obtain the image of each element of C under the modular and extended modular group by continued fractions.
📊
Key Findings & Data:The modular group takes the upper half of the complex plane into itself.
💡
Conclusion & Impact:Using these images, unlike known methods, we can show that the extended modular group takes the upper half of the complex plane into H.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Modular group · Extended modular group · Continued fraction

Abstract

In this paper we obtain the image of each element of C under the modular and extended modular group by continued fractions. The modular group takes the upper half of the complex plane into itself. Using these images, unlike known methods, we can show that the extended modular group takes the upper half of the complex plane into H.

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Cite This Article

KORUOĞLU, Ö. (2022). THE IMAGES UNDER THE MODULAR GROUP AND EXTENDED MODULAR GROUP. Hacettepe Journal of Mathematics and Statistics, Cilt: 40 - Sayı: 1, 15–20. https://ajindex.com/articles/530569

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References (11)

  1. Bizim, O. The extended modular group, Hacettepe Bull. Nat. Sci. Eng. Ser. B 26, 1–11, 1997.
  2. Fine, B. Trace Classes and quadratic forms in the modular group, Canad. Math. Bull. 37 (2), 202–212, 1994. [3] ˙Ikikardes, S., Koruo˘glu, ¨O, Sahin, R. and Cangul, I. N. One relator quotients of the extended modular group, Adv. Stud. Contemp. Math. (Kyungshang) 17 (2), 203–210, 2008.
  3. Jones, Gareth A. and Singerman, D. Complex Functions, an algebraic and geometric view- point (Cambridge University Press, Cambridge, 1987).
  4. Jones, G. A. and Thornton, J. S. Automorphisms and congruence subgroups of the extended modular group, J. London Math. Soc. 34 (2), 26–40, 1986. Koruo˘glu, ¨O., S¸ahin, R. and ˙Ikikarde¸s, S. Trace classes and fixed points for the extended modular group Γ, Turkish J. Math. 32, 11–19, 2008. Koruo˘glu, ¨O. The determination of the parabolic points in the modular group and the ex- tended modular group by continued fractions, Bulletin of the Malaysian Mathematical Sci- ences Society 33 (3), 439–445, 2010.
  5. Lang, M. L. Normalisers of subgroups of the modular group, J. Algebra 248 (1), 202–218, 2002.
  6. Mushtaq, Q. and Hayat, U. Horadam generalized Fibonacci numbers and the modular group, Indian J. Pure Appl. Math. 38 (5), 345–352, 2007.
  7. Mushtaq, Q. and Hayat, U. Pell numbers, Pell-Lucas numbers and modular group, Algebra Colloq. 14 (1), 97–102, 2007.
  8. Newman, M. The structure of some subgroups of the modular group, Illinois J. Math. 6, 480–487, 1962. [12] Newman, M. Free subgroups and normal subgroups of the modular group, Illinois J. Math. 8, 262–265, 1964.
  9. Rosen, D. A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J. 21, 549–564, 1954.
  10. Sahin, R., ˙Ikikardes S. and Koruo˘glu, ¨O. On the power subgroups of the extended modular group Γ, Turkish J. Math. 29, 143–151, 2004.
  11. Singerman, D. PSL(2,q) as an image of the extended modular group with applications to group actions on surfaces, Proc. Edinb. Math. Soc. II. Ser. 30, 143–151, 1987.