Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪...

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Main Authors:	Tianlan Chen, Muhammad Nadeem Bari, Muhammad Aslam Malik, Hafiz Muhammad Afzal Siddiqui, Jia-Bao Liu
Format:	Article
Language:	English
Published:	Hindawi Limited 2020-01-01
Series:	Journal of Mathematics
Online Access:	http://dx.doi.org/10.1155/2020/9568254

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spelling	doaj-c6b233922f8b4813b34fb530efa0d97a2020-11-25T03:11:35ZengHindawi LimitedJournal of Mathematics2314-46292314-47852020-01-01202010.1155/2020/95682549568254Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic FieldsTianlan Chen0Muhammad Nadeem Bari1Muhammad Aslam Malik2Hafiz Muhammad Afzal Siddiqui3Jia-Bao Liu4Practice Training Center for Engineering Technology Talents of Guizhou Minzu University, Guiyang, Guizhou 550025, ChinaDepartment of Mathematics, University of the Punjab, Lahore 54590, PakistanDepartment of Mathematics, University of the Punjab, Lahore 54590, PakistanDepartment of Mathematics, COMSATS University Islamabad, Lahore-Campus 54000, PakistanSchool of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, ChinaReduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.http://dx.doi.org/10.1155/2020/9568254
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Tianlan Chen Muhammad Nadeem Bari Muhammad Aslam Malik Hafiz Muhammad Afzal Siddiqui Jia-Bao Liu
spellingShingle	Tianlan Chen Muhammad Nadeem Bari Muhammad Aslam Malik Hafiz Muhammad Afzal Siddiqui Jia-Bao Liu Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields Journal of Mathematics
author_facet	Tianlan Chen Muhammad Nadeem Bari Muhammad Aslam Malik Hafiz Muhammad Afzal Siddiqui Jia-Bao Liu
author_sort	Tianlan Chen
title	Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields
title_short	Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields
title_full	Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields
title_fullStr	Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields
title_full_unstemmed	Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields
title_sort	icosahedral group and classification of psl(2, z)-orbits of real quadratic fields
publisher	Hindawi Limited
series	Journal of Mathematics
issn	2314-4629 2314-4785
publishDate	2020-01-01
description	Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.
url	http://dx.doi.org/10.1155/2020/9568254
work_keys_str_mv	AT tianlanchen icosahedralgroupandclassificationofpsl2zorbitsofrealquadraticfields AT muhammadnadeembari icosahedralgroupandclassificationofpsl2zorbitsofrealquadraticfields AT muhammadaslammalik icosahedralgroupandclassificationofpsl2zorbitsofrealquadraticfields AT hafizmuhammadafzalsiddiqui icosahedralgroupandclassificationofpsl2zorbitsofrealquadraticfields AT jiabaoliu icosahedralgroupandclassificationofpsl2zorbitsofrealquadraticfields
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Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields

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