Groups containing locally maximal product-free sets of size 4
Every locally maximal product-free set \(S\) in a finite group \(G\) satisfies \(G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy\mid x,y\in S\}\), \(S^{-1}S=\{x^{-1}y\mid x,y\in S\}\), \(SS^{-1}=\{xy^{-1}\mid x,y\in S\}\) and \(\sqrt{S}=\{x\in G\mid x^2\in S\}\). To better under...
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| Дата: | 2021 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2021
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-13472021-07-19T08:39:30Z Groups containing locally maximal product-free sets of size 4 Anabanti, C. S. product-free sets, locally maximal, maximal, groups 20D60, 05E15, 11B75 Every locally maximal product-free set \(S\) in a finite group \(G\) satisfies \(G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy\mid x,y\in S\}\), \(S^{-1}S=\{x^{-1}y\mid x,y\in S\}\), \(SS^{-1}=\{xy^{-1}\mid x,y\in S\}\) and \(\sqrt{S}=\{x\in G\mid x^2\in S\}\). To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set \(S\) in a finite abelian group satisfy \(|\sqrt{S}|\leq 2|S|\). This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size \(4\), continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size \(4\), and conclude with a conjecture on the size \(4\) problem as well as an open problem on the general case. Lugansk National Taras Shevchenko University 2021-07-19 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1347 10.12958/adm1347 Algebra and Discrete Mathematics; Vol 31, No 2 (2021) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1347/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1347/505 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1347/866 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1347/867 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1347/868 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1347/869 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1347/870 Copyright (c) 2021 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
product-free sets locally maximal maximal groups 20D60 05E15 11B75 |
| spellingShingle |
product-free sets locally maximal maximal groups 20D60 05E15 11B75 Anabanti, C. S. Groups containing locally maximal product-free sets of size 4 |
| topic_facet |
product-free sets locally maximal maximal groups 20D60 05E15 11B75 |
| format |
Article |
| author |
Anabanti, C. S. |
| author_facet |
Anabanti, C. S. |
| author_sort |
Anabanti, C. S. |
| title |
Groups containing locally maximal product-free sets of size 4 |
| title_short |
Groups containing locally maximal product-free sets of size 4 |
| title_full |
Groups containing locally maximal product-free sets of size 4 |
| title_fullStr |
Groups containing locally maximal product-free sets of size 4 |
| title_full_unstemmed |
Groups containing locally maximal product-free sets of size 4 |
| title_sort |
groups containing locally maximal product-free sets of size 4 |
| description |
Every locally maximal product-free set \(S\) in a finite group \(G\) satisfies \(G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy\mid x,y\in S\}\), \(S^{-1}S=\{x^{-1}y\mid x,y\in S\}\), \(SS^{-1}=\{xy^{-1}\mid x,y\in S\}\) and \(\sqrt{S}=\{x\in G\mid x^2\in S\}\). To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set \(S\) in a finite abelian group satisfy \(|\sqrt{S}|\leq 2|S|\). This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size \(4\), continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size \(4\), and conclude with a conjecture on the size \(4\) problem as well as an open problem on the general case. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2021 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1347 |
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2024-04-12T06:25:37Z |
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