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Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups

Year 2020, Volume: 3 Issue: 1, 110 - 114, 15.12.2020

İbrahim Gökcan Ali Hikmet Değer

Abstract

In [3], the modular group, the movement of an element of the modular group on Q ̂ (extended set of rational numbers) in hyperbolic geometry, and Farey graph, G_(u,n) and F_(u,n) were investigated. Furthermore, it is indicated that the fixed of any two points is conjugated in Γ, and the element of the modular group that leaves constant an element on Q ̂ is infinite period. Hence, the element of the modular group that leaves the ∞ element constant is symbolized as Γ_∞. In the same study, H, the subgroups of Γ of containing Γ_∞ are obtained and its invariant equivalence relations are generated on Q ̂. Taking these points into account, in this study, we show that, the element that fixed x/y in modular group based on the choice of x/y for x,y∈Z and (x,y)=1, instead of a special value of set Q ̂, such as ∞. Similarly, we study subgroups H containing Γ_(x/y) and we examine its the invariants equivalence relations on Q ̂.

Keywords

Infinite period Invariant equivalence relations Modular group

References

  • 1 B. Schoeneberg, Elliptic Modular Functions , Springer-Verlag, Berlin, Heidelberg, New York,(1974).
  • 2 C.C. Sims, Graphs and Finite Permutation Groups, Math. Z. 95 (1967), 75-86.
  • 3 G.A. Jones,D. Singerman and K. Wicks, The Modular Group and Generalized Farey Graphs, London Math. Soc. Lecture Notes, CUP, Cambridge, 160 (1991), 316-338.
  • 4 N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures , London Math. Soc. Lecture Notes 33, Cambridge University Press, Cambridge, (1979).
  • 5 M. Akbas, On Suborbital Graphs for The Modular Group, Bull. London Math. Soc., 33 (2001), 647-652.

Year 2020, Volume: 3 Issue: 1, 110 - 114, 15.12.2020

İbrahim Gökcan Ali Hikmet Değer

Abstract

References

  • 1 B. Schoeneberg, Elliptic Modular Functions , Springer-Verlag, Berlin, Heidelberg, New York,(1974).
  • 2 C.C. Sims, Graphs and Finite Permutation Groups, Math. Z. 95 (1967), 75-86.
  • 3 G.A. Jones,D. Singerman and K. Wicks, The Modular Group and Generalized Farey Graphs, London Math. Soc. Lecture Notes, CUP, Cambridge, 160 (1991), 316-338.
  • 4 N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures , London Math. Soc. Lecture Notes 33, Cambridge University Press, Cambridge, (1979).
  • 5 M. Akbas, On Suborbital Graphs for The Modular Group, Bull. London Math. Soc., 33 (2001), 647-652.
There are 5 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

İbrahim Gökcan 0000-0002-6933-8494

Ali Hikmet Değer 0000-0003-0764-715X

Publication Date December 15, 2020
Acceptance Date September 28, 2020
Published in Issue Year 2020 Volume: 3 Issue: 1

Cite

APA Gökcan, İ., & Değer, A. H. (2020). Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups. Conference Proceedings of Science and Technology, 3(1), 110-114.
AMA Gökcan İ, Değer AH. Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups. Conference Proceedings of Science and Technology. December 2020;3(1):110-114.
Chicago Gökcan, İbrahim, and Ali Hikmet Değer. “Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups”. Conference Proceedings of Science and Technology 3, no. 1 (December 2020): 110-14.
EndNote Gökcan İ, Değer AH (December 1, 2020) Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups. Conference Proceedings of Science and Technology 3 1 110–114.
IEEE İ. Gökcan and A. H. Değer, “Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups”, Conference Proceedings of Science and Technology, vol. 3, no. 1, pp. 110–114, 2020.
ISNAD Gökcan, İbrahim - Değer, Ali Hikmet. “Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups”. Conference Proceedings of Science and Technology 3/1 (December 2020), 110-114.
JAMA Gökcan İ, Değer AH. Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups. Conference Proceedings of Science and Technology. 2020;3:110–114.
MLA Gökcan, İbrahim and Ali Hikmet Değer. “Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups”. Conference Proceedings of Science and Technology, vol. 3, no. 1, 2020, pp. 110-4.
Vancouver Gökcan İ, Değer AH. Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups. Conference Proceedings of Science and Technology. 2020;3(1):110-4.
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