On the structure of finite groups associated to regular non-centralizer graphs
The non-centralizer graph of a finite group $ G $ is the simple graph $ \Upsilon_G $ whose vertices are the elements of $ G $ with two vertices are adjacent if their centralizers are distinct. The induced non-centralizer graph of $ G $ is the induced subgraph of $ \Upsilon_G $ on $ G\setminus Z(G) $...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-11-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231585?viewType=HTML |
| _version_ | 1797405565490561024 |
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| author | Tariq A. Alraqad Hicham Saber |
| author_facet | Tariq A. Alraqad Hicham Saber |
| author_sort | Tariq A. Alraqad |
| collection | DOAJ |
| description | The non-centralizer graph of a finite group $ G $ is the simple graph $ \Upsilon_G $ whose vertices are the elements of $ G $ with two vertices are adjacent if their centralizers are distinct. The induced non-centralizer graph of $ G $ is the induced subgraph of $ \Upsilon_G $ on $ G\setminus Z(G) $. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups and induced regular groups. We prove that if a group $ G $ is regular, then $ G/Z(G) $ as an elementary $ 2 $-group. Using the concept of maximal centralizers, we succeeded in proving that if $ G $ is induced regular, then $ G/Z(G) $ is a $ p $-group. We also show that a group $ G $ is induced regular if and only if it is the direct product of an induced regular $ p $-group and an abelian group. |
| first_indexed | 2024-03-09T03:11:51Z |
| format | Article |
| id | doaj.art-58645f6fbbc1499d9595f9eadb5a196f |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-09T03:11:51Z |
| publishDate | 2023-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-58645f6fbbc1499d9595f9eadb5a196f2023-12-04T01:25:32ZengAIMS PressAIMS Mathematics2473-69882023-11-01812309813099110.3934/math.20231585On the structure of finite groups associated to regular non-centralizer graphsTariq A. Alraqad 0Hicham Saber1Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi ArabiaDepartment of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi ArabiaThe non-centralizer graph of a finite group $ G $ is the simple graph $ \Upsilon_G $ whose vertices are the elements of $ G $ with two vertices are adjacent if their centralizers are distinct. The induced non-centralizer graph of $ G $ is the induced subgraph of $ \Upsilon_G $ on $ G\setminus Z(G) $. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups and induced regular groups. We prove that if a group $ G $ is regular, then $ G/Z(G) $ as an elementary $ 2 $-group. Using the concept of maximal centralizers, we succeeded in proving that if $ G $ is induced regular, then $ G/Z(G) $ is a $ p $-group. We also show that a group $ G $ is induced regular if and only if it is the direct product of an induced regular $ p $-group and an abelian group. https://www.aimspress.com/article/doi/10.3934/math.20231585?viewType=HTMLcentralizersfinite groupsgraphregular |
| spellingShingle | Tariq A. Alraqad Hicham Saber On the structure of finite groups associated to regular non-centralizer graphs AIMS Mathematics centralizers finite groups graph regular |
| title | On the structure of finite groups associated to regular non-centralizer graphs |
| title_full | On the structure of finite groups associated to regular non-centralizer graphs |
| title_fullStr | On the structure of finite groups associated to regular non-centralizer graphs |
| title_full_unstemmed | On the structure of finite groups associated to regular non-centralizer graphs |
| title_short | On the structure of finite groups associated to regular non-centralizer graphs |
| title_sort | on the structure of finite groups associated to regular non centralizer graphs |
| topic | centralizers finite groups graph regular |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231585?viewType=HTML |
| work_keys_str_mv | AT tariqaalraqad onthestructureoffinitegroupsassociatedtoregularnoncentralizergraphs AT hichamsaber onthestructureoffinitegroupsassociatedtoregularnoncentralizergraphs |