A new approach to character-free proof for Frobenius theorem
Let G be a Frobenius group. Using character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this note, we prove it, by assuming that Frobenius groups are non-si...
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| Format: | Article |
| Language: | English |
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Amirkabir University of Technology
2023-02-01
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| Series: | AUT Journal of Mathematics and Computing |
| Subjects: | |
| Online Access: | https://ajmc.aut.ac.ir/article_4863_36ef014a4f85c1bbee9f6f358b602a34.pdf |
| _version_ | 1797307091888635904 |
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| author | Seyedeh Fatemeh Arfaeezarandi Vahid Shahverdi |
| author_facet | Seyedeh Fatemeh Arfaeezarandi Vahid Shahverdi |
| author_sort | Seyedeh Fatemeh Arfaeezarandi |
| collection | DOAJ |
| description | Let G be a Frobenius group. Using character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this note, we prove it, by assuming that Frobenius groups are non-simple. Also, we prove that whether K is a subgroup of G or not, Sylow 2-subgroups of G are either cyclic or generalized quaternion group. Also by assuming some additional arithmetical hypothesis on G we prove Frobenius theorem. We should mention that our proof is character-free. |
| first_indexed | 2024-03-08T00:52:19Z |
| format | Article |
| id | doaj.art-cf49cabdd83c43d6a6a883556a685ad4 |
| institution | Directory Open Access Journal |
| issn | 2783-2449 2783-2287 |
| language | English |
| last_indexed | 2024-03-08T00:52:19Z |
| publishDate | 2023-02-01 |
| publisher | Amirkabir University of Technology |
| record_format | Article |
| series | AUT Journal of Mathematics and Computing |
| spelling | doaj.art-cf49cabdd83c43d6a6a883556a685ad42024-02-14T19:39:06ZengAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24492783-22872023-02-01419910310.22060/ajmc.2022.21305.10854863A new approach to character-free proof for Frobenius theoremSeyedeh Fatemeh Arfaeezarandi0Vahid Shahverdi1Department of Mathematics, Stony Brook University, Stony Brook, New York, USADepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, SwedenLet G be a Frobenius group. Using character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this note, we prove it, by assuming that Frobenius groups are non-simple. Also, we prove that whether K is a subgroup of G or not, Sylow 2-subgroups of G are either cyclic or generalized quaternion group. Also by assuming some additional arithmetical hypothesis on G we prove Frobenius theorem. We should mention that our proof is character-free.https://ajmc.aut.ac.ir/article_4863_36ef014a4f85c1bbee9f6f358b602a34.pdffinite groupfrobenius groupfrobenius theorem |
| spellingShingle | Seyedeh Fatemeh Arfaeezarandi Vahid Shahverdi A new approach to character-free proof for Frobenius theorem AUT Journal of Mathematics and Computing finite group frobenius group frobenius theorem |
| title | A new approach to character-free proof for Frobenius theorem |
| title_full | A new approach to character-free proof for Frobenius theorem |
| title_fullStr | A new approach to character-free proof for Frobenius theorem |
| title_full_unstemmed | A new approach to character-free proof for Frobenius theorem |
| title_short | A new approach to character-free proof for Frobenius theorem |
| title_sort | new approach to character free proof for frobenius theorem |
| topic | finite group frobenius group frobenius theorem |
| url | https://ajmc.aut.ac.ir/article_4863_36ef014a4f85c1bbee9f6f358b602a34.pdf |
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