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Journals / Turkish Journal of Mathematics / 2010 / Cilt: 34 - Sayı: 2

Finite subquandles of sphere

Nülifer ÖZDEMİR
Hüseyin AZCAN
Pages
293–303
Publication Date
2022-11-05
DOI
—
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AI Summary & Key Takeaways

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

It is proved that the finite subquandles Q1 and Q2 of sphere whose all elements are not on the same great circle are isomorphic if and only if the subgroups GQ1 and GQ2 of O(3) are isomorphic by which finite subquandles of sphere are classified.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this work finite subquandles of sphere are classified by using classification of subgroups of orthogonal group O(3).
📊
Key Findings & Data:For any subquandle Q of sphere there is a subgroup GQ of O(3) associated with Q.
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Conclusion & Impact:It is proved that the finite subquandles Q1 and Q2 of sphere whose all elements are not on the same great circle are isomorphic if and only if the subgroups GQ1 and GQ2 of O(3) are isomorphic by which finite subquandles of sphere are classified.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Quandle · orthogonal group. · Quandle

Abstract

In this work finite subquandles of sphere are classified by using classification of subgroups of orthogonal group O(3). For any subquandle Q of sphere there is a subgroup GQ of O(3) associated with Q. It is shown that if Q is a finite (infinite) subquandle, then GQ is a finite (infinite) subgroup. Finite subquandles of sphere are obtained from actions of finite subgroups of SO(3) on sphere. It is proved that the finite subquandles Q1 and Q2 of sphere whose all elements are not on the same great circle are isomorphic if and only if the subgroups GQ1 and GQ2 of O(3) are isomorphic by which finite subquandles of sphere are classified.

Özet

In this work finite subquandles of sphere are classified by using classification of subgroups of orthogonal group O(3). For any subquandle Q of sphere there is a subgroup GQ of O(3) associated with Q. It is shown that if Q is a finite (infinite) subquandle, then GQ is a finite (infinite) subgroup. Finite subquandles of sphere are obtained from actions of finite subgroups of SO(3) on sphere. It is proved that the finite subquandles Q1 and Q2 of sphere whose all elements are not on the same great circle are isomorphic if and only if the subgroups GQ1 and GQ2 of O(3) are isomorphic by which finite subquandles of sphere are classified.

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Cite This Article

ÖZDEMİR, N., AZCAN, H. (2022). Finite subquandles of sphere. Turkish Journal of Mathematics, Cilt: 34 - Sayı: 2, 293–303. https://ajindex.com/articles/608526

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References (6)

  1. intersection point. This is the point ylwe are looking for. As a result Φ(xl) = is deŞned as yl. Continuing like this we get the isomorphism Φ. Hence Q1∼ y1, y2,· · ·, yi,· · ·, yn ∼ σy1, σy2,· · ·, σyn ∼ = Q2is obtained. Note that the group = y1, σy2,· · ·, σi· · ·, σyn. For n≥ 3, Şnite subgroups of O(n) that is generated by reflections is given in [6]. Giving explicit Şnite subquandles of Snby using the Şnite reflction subgroups of O(n + 1) may be the subject of another work. Brieskorn, E.: Automorphic sets and braids and singularities, Contemporary Mathematics, 78, 45-115 (1988).
  2. Joyce, D.: A classifying invariant of knots, the knot quandle, Journal of Pure and Applied Algebra, 23, 37-65 (1982).
  3. Roger, F. and Rourke, C.: Racks and links in codimension two, Journal of Knot Theory and Its RamiŞcations, 1, 406 (1992).
  4. Armstrong, M.A.: Groups and symmetry, Springer-Verlag (1988).
  5. Beardon, A.F.: The geometry of discrete groups, Springer-Verlag (1983).
  6. Grove, L.C., Benson, C.T.: Finite reflection groups, Springer-Verlag (1985). N¨ulifer ¨OZDEM˙IR and H¨useyin AZCAN Department of Mathematics, Science Faculty, Anadolu University,, Eski¸sehir, TURKEY e-mail: [email protected] e-mail: [email protected]