On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures
The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups G1×G2, and it inv...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/7705500 |
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| author | Fanning Meng Junhui Guo |
| author_facet | Fanning Meng Junhui Guo |
| author_sort | Fanning Meng |
| collection | DOAJ |
| description | The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups G1×G2, and it involves isomorphisms between quotient groups of subgroups of G1 and G2. In this paper, we first extend Goursat’s lemma to R-algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, R-modules, and R-algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, R-modules, and R-algebras, respectively. |
| format | Article |
| id | doaj-art-3fb91d4d7cb440dd88441839f5b07450 |
| institution | OA Journals |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-3fb91d4d7cb440dd88441839f5b074502025-08-20T02:08:04ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/7705500On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic StructuresFanning Meng0Junhui Guo1School of Mathematics and Information ScienceSchool of Mathematics and Information ScienceThe Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups G1×G2, and it involves isomorphisms between quotient groups of subgroups of G1 and G2. In this paper, we first extend Goursat’s lemma to R-algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, R-modules, and R-algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, R-modules, and R-algebras, respectively.http://dx.doi.org/10.1155/2022/7705500 |
| spellingShingle | Fanning Meng Junhui Guo On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures Journal of Mathematics |
| title | On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures |
| title_full | On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures |
| title_fullStr | On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures |
| title_full_unstemmed | On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures |
| title_short | On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures |
| title_sort | on the extensions of zassenhaus lemma and goursat s lemma to algebraic structures |
| url | http://dx.doi.org/10.1155/2022/7705500 |
| work_keys_str_mv | AT fanningmeng ontheextensionsofzassenhauslemmaandgoursatslemmatoalgebraicstructures AT junhuiguo ontheextensionsofzassenhauslemmaandgoursatslemmatoalgebraicstructures |