A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms

The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article...
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Gespeichert in:

Autor*in:	Abdul Razaq [verfasserIn] Ghaliah Alhamzi [verfasserIn] Asima Razzaque [verfasserIn] Harish Garg [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism

Übergeordnetes Werk:	In: Symmetry - MDPI AG, 2009, 14(2022), 10, p 2084
Übergeordnetes Werk:	volume:14 ; year:2022 ; number:10, p 2084

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/sym14102084

Katalog-ID:	DOAJ027402797

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520			\|a The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved.
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allfields	10.3390/sym14102084 doi (DE-627)DOAJ027402797 (DE-599)DOAJ5f7b7506eae04d43961de7d906e713aa DE-627 ger DE-627 rakwb eng QA1-939 Abdul Razaq verfasserin aut A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved. Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism Mathematics Ghaliah Alhamzi verfasserin aut Asima Razzaque verfasserin aut Harish Garg verfasserin aut In Symmetry MDPI AG, 2009 14(2022), 10, p 2084 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:14 year:2022 number:10, p 2084 https://doi.org/10.3390/sym14102084 kostenfrei https://doaj.org/article/5f7b7506eae04d43961de7d906e713aa kostenfrei https://www.mdpi.com/2073-8994/14/10/2084 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 14 2022 10, p 2084
spelling	10.3390/sym14102084 doi (DE-627)DOAJ027402797 (DE-599)DOAJ5f7b7506eae04d43961de7d906e713aa DE-627 ger DE-627 rakwb eng QA1-939 Abdul Razaq verfasserin aut A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved. Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism Mathematics Ghaliah Alhamzi verfasserin aut Asima Razzaque verfasserin aut Harish Garg verfasserin aut In Symmetry MDPI AG, 2009 14(2022), 10, p 2084 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:14 year:2022 number:10, p 2084 https://doi.org/10.3390/sym14102084 kostenfrei https://doaj.org/article/5f7b7506eae04d43961de7d906e713aa kostenfrei https://www.mdpi.com/2073-8994/14/10/2084 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 14 2022 10, p 2084
allfields_unstemmed	10.3390/sym14102084 doi (DE-627)DOAJ027402797 (DE-599)DOAJ5f7b7506eae04d43961de7d906e713aa DE-627 ger DE-627 rakwb eng QA1-939 Abdul Razaq verfasserin aut A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved. Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism Mathematics Ghaliah Alhamzi verfasserin aut Asima Razzaque verfasserin aut Harish Garg verfasserin aut In Symmetry MDPI AG, 2009 14(2022), 10, p 2084 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:14 year:2022 number:10, p 2084 https://doi.org/10.3390/sym14102084 kostenfrei https://doaj.org/article/5f7b7506eae04d43961de7d906e713aa kostenfrei https://www.mdpi.com/2073-8994/14/10/2084 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 14 2022 10, p 2084
allfieldsGer	10.3390/sym14102084 doi (DE-627)DOAJ027402797 (DE-599)DOAJ5f7b7506eae04d43961de7d906e713aa DE-627 ger DE-627 rakwb eng QA1-939 Abdul Razaq verfasserin aut A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved. Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism Mathematics Ghaliah Alhamzi verfasserin aut Asima Razzaque verfasserin aut Harish Garg verfasserin aut In Symmetry MDPI AG, 2009 14(2022), 10, p 2084 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:14 year:2022 number:10, p 2084 https://doi.org/10.3390/sym14102084 kostenfrei https://doaj.org/article/5f7b7506eae04d43961de7d906e713aa kostenfrei https://www.mdpi.com/2073-8994/14/10/2084 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 14 2022 10, p 2084
allfieldsSound	10.3390/sym14102084 doi (DE-627)DOAJ027402797 (DE-599)DOAJ5f7b7506eae04d43961de7d906e713aa DE-627 ger DE-627 rakwb eng QA1-939 Abdul Razaq verfasserin aut A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved. Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism Mathematics Ghaliah Alhamzi verfasserin aut Asima Razzaque verfasserin aut Harish Garg verfasserin aut In Symmetry MDPI AG, 2009 14(2022), 10, p 2084 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:14 year:2022 number:10, p 2084 https://doi.org/10.3390/sym14102084 kostenfrei https://doaj.org/article/5f7b7506eae04d43961de7d906e713aa kostenfrei https://www.mdpi.com/2073-8994/14/10/2084 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 14 2022 10, p 2084
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spellingShingle	Abdul Razaq misc QA1-939 misc Pythagorean fuzzy set misc Pythagorean fuzzy subgroup misc Pythagorean fuzzy normal subgroup misc Pythagorean fuzzy isomorphism misc Mathematics A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms
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topic_title	QA1-939 A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms Pythagorean fuzzy set Pythagorean fuzzy subgroup Pythagorean fuzzy normal subgroup Pythagorean fuzzy isomorphism
topic	misc QA1-939 misc Pythagorean fuzzy set misc Pythagorean fuzzy subgroup misc Pythagorean fuzzy normal subgroup misc Pythagorean fuzzy isomorphism misc Mathematics
topic_unstemmed	misc QA1-939 misc Pythagorean fuzzy set misc Pythagorean fuzzy subgroup misc Pythagorean fuzzy normal subgroup misc Pythagorean fuzzy isomorphism misc Mathematics
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title	A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms
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title_full	A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms
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title_sort	comprehensive study on pythagorean fuzzy normal subgroups and pythagorean fuzzy isomorphisms
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title_auth	A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms
abstract	The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved.
abstractGer	The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved.
abstract_unstemmed	The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved.
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title_short	A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms
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We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<W</mi<</semantics<</math<</inline-formula< relative to its normal subgroup <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mi<S</mi<</semantics<</math<</inline-formula< by defining a PFSG of <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mfrac bevelled="true"<<mi<W</mi<<mi<S</mi<</mfrac<</mrow<</semantics<</math<</inline-formula<. 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A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms

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