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 ∪ SS ∪ S⁻¹S ∪ SS⁻¹ ∪ √S, where SS = {xy | x, y ∈ S}, S⁻¹S = {x⁻¹y | x, y ∈ S}, SS⁻¹ = {xy⁻¹ | x, y ∈ S} and √S = {x ∈ G | x² ∈ S}. To better understand locally maximal product-free sets, Bertram asked whether every locally...
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| Дата: | 2021 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2021
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/188705 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Groups containing locally maximal product-free sets of size 4 / C.S. Anabanti // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 167–194. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-188705 |
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irk-123456789-1887052023-03-12T01:28:56Z Groups containing locally maximal product-free sets of size 4 Anabanti, C.S. Every locally maximal product-free set S in a finite group G satisfies G = S ∪ SS ∪ S⁻¹S ∪ SS⁻¹ ∪ √S, where SS = {xy | x, y ∈ S}, S⁻¹S = {x⁻¹y | x, y ∈ S}, SS⁻¹ = {xy⁻¹ | x, y ∈ S} and √S = {x ∈ G | x² ∈ 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 |√S| ≤ 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. 2021 Article Groups containing locally maximal product-free sets of size 4 / C.S. Anabanti // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 167–194. — Бібліогр.: 12 назв. — англ. 1726-3255 DOI:10.12958/adm1347 2020 MSC: 20D60, 05E15, 11B75 http://dspace.nbuv.gov.ua/handle/123456789/188705 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Every locally maximal product-free set S in a finite group G satisfies G = S ∪ SS ∪ S⁻¹S ∪ SS⁻¹ ∪ √S, where SS = {xy | x, y ∈ S}, S⁻¹S = {x⁻¹y | x, y ∈ S}, SS⁻¹ = {xy⁻¹ | x, y ∈ S} and √S = {x ∈ G | x² ∈ 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 |√S| ≤ 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. |
| format |
Article |
| author |
Anabanti, C.S. |
| spellingShingle |
Anabanti, C.S. Groups containing locally maximal product-free sets of size 4 Algebra and Discrete Mathematics |
| 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 |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2021 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188705 |
| citation_txt |
Groups containing locally maximal product-free sets of size 4 / C.S. Anabanti // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 167–194. — Бібліогр.: 12 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT anabantics groupscontaininglocallymaximalproductfreesetsofsize4 |
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2023-10-18T23:08:58Z |
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