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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| Опубліковано в: : | Algebra and Discrete Mathematics |
|---|---|
| Дата: | 2021 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2021
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| Онлайн доступ: | https://nasplib.isofts.kiev.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| _version_ | 1862580972836356096 |
|---|---|
| author | Anabanti, C.S. |
| author_facet | Anabanti, C.S. |
| 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. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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.
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| first_indexed | 2025-11-26T21:44:28Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-188705 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-26T21:44:28Z |
| publishDate | 2021 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Anabanti, C.S. 2023-03-11T15:51:12Z 2023-03-11T15:51:12Z 2021 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 https://nasplib.isofts.kiev.ua/handle/123456789/188705 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. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Groups containing locally maximal product-free sets of size 4 Article published earlier |
| spellingShingle | Groups containing locally maximal product-free sets of size 4 Anabanti, C.S. |
| title | 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_short | Groups containing locally maximal product-free sets of size 4 |
| title_sort | groups containing locally maximal product-free sets of size 4 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188705 |
| work_keys_str_mv | AT anabantics groupscontaininglocallymaximalproductfreesetsofsize4 |