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Journals / Conference Proceedings of Science and Technology / 2026 / Cilt: 3 Sayı: 1

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

İbrahim Gökcan(KARADENİZ TEKNİK ÜNİVERSİTESİ)0000-0002-6933-8494
Ali Hikmet Değer(KARADENIZ TECHNICAL UNIVERSITY, FACULTY OF SCIENCE)0000-0003-0764-715X
Pages
110–114
Publication Date
2020-12-15
DOI
—
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AI Summary & Key Takeaways

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

Similarly, we study subgroups H containing Γ_(x/y) and we examine its the invariants equivalence relations on Q ̂.

📌 3 Key Takeaways

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Objective & Hypothesis: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.
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Key Findings & Data:In the same study, H, the subgroups of Γ of containing Γ_∞ are obtained and its invariant equivalence relations are generated on Q ̂.
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Conclusion & Impact:Similarly, we study subgroups H containing Γ_(x/y) and we examine its the invariants equivalence relations on Q ̂.
🔬Design:Konferans Bildirisi
📍Scope:Comprehensive Sample
⚙️Variables:Infinite period · Invariant equivalence relations · Modular group

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 ̂.

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Cite This Article

Gökcan, İ., Değer, A. H. (2020). Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups. Conference Proceedings of Science and Technology, Cilt: 3 Sayı: 1, 110–114. https://ajindex.com/articles/1648717

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References (7)

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