$Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings$

Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings

The paper is devoted to the study of local nearrings (those with identity, for which all non-invertible elements form subgroups of their additive group). A study of such nearrings was first initiated by C. J. Maxson in 1968, and the problem on the determination of the finite $p$-groups, which are...

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Дата:	2024
Автори:	Raievska, Iryna, Raievska, Maryna
Формат:	Стаття
Мова:	English
Опубліковано:	Lugansk National Taras Shevchenko University 2024
Теми:	local nearring $p$-group nilpotency class $2$ 16Y30
Онлайн доступ:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2269
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

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spelling	oai:ojs.admjournal.luguniv.edu.ua:article-22692024-09-23T09:29:11Z Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings Raievska, Iryna Raievska, Maryna local nearring, $p$-group, nilpotency class $2$ 16Y30 The paper is devoted to the study of local nearrings (those with identity, for which all non-invertible elements form subgroups of their additive group). A study of such nearrings was first initiated by C. J. Maxson in 1968, and the problem on the determination of the finite $p$-groups, which are the additive groups of local nearrings have become one of the most important. Particular cases of this (still unsolved) problem have been studied in many works. In previous papers the authors have shown that, up to isomorphism, there exist at least $p$ local nearrings on elementary abelian additive groups of order $p^3$, which are not nearfields, and at least $p+1$ on each non-metacyclic non-abelian or metacyclic abelian groups of order $p^3$. In this paper we study the groups of nilpotency class $2$ of order $p^4$, which are the additive groups of local nearrings. It is proved that, for odd $p$, $4$ out of total number $6$ of such groups are the additive groups of local nearrings. Explicit examples of the corresponding local nearrings are provided. Lugansk National Taras Shevchenko University 2024-09-23 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2269 10.12958/adm2269 Algebra and Discrete Mathematics; Vol 38, No 1 (2024) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2269/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2269/1193 Copyright (c) 2024 Algebra and Discrete Mathematics
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2024-09-23T09:29:11Z
collection	OJS
language	English
topic	local nearring $p$-group nilpotency class $2$ 16Y30
spellingShingle	local nearring $p$-group nilpotency class $2$ 16Y30 Raievska, Iryna Raievska, Maryna Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
topic_facet	local nearring $p$-group nilpotency class $2$ 16Y30
format	Article
author	Raievska, Iryna Raievska, Maryna
author_facet	Raievska, Iryna Raievska, Maryna
author_sort	Raievska, Iryna
title	Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
title_short	Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
title_full	Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
title_fullStr	Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
title_full_unstemmed	Groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
title_sort	groups of nilpotency class $2$ of order $p^4$ as additive groups of local nearrings
description	The paper is devoted to the study of local nearrings (those with identity, for which all non-invertible elements form subgroups of their additive group). A study of such nearrings was first initiated by C. J. Maxson in 1968, and the problem on the determination of the finite $p$-groups, which are the additive groups of local nearrings have become one of the most important. Particular cases of this (still unsolved) problem have been studied in many works. In previous papers the authors have shown that, up to isomorphism, there exist at least $p$ local nearrings on elementary abelian additive groups of order $p^3$, which are not nearfields, and at least $p+1$ on each non-metacyclic non-abelian or metacyclic abelian groups of order $p^3$. In this paper we study the groups of nilpotency class $2$ of order $p^4$, which are the additive groups of local nearrings. It is proved that, for odd $p$, $4$ out of total number $6$ of such groups are the additive groups of local nearrings. Explicit examples of the corresponding local nearrings are provided.
publisher	Lugansk National Taras Shevchenko University
publishDate	2024
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2269
work_keys_str_mv	AT raievskairyna groupsofnilpotencyclass2oforderp4asadditivegroupsoflocalnearrings AT raievskamaryna groupsofnilpotencyclass2oforderp4asadditivegroupsoflocalnearrings
first_indexed	2024-09-24T04:03:45Z
last_indexed	2024-09-24T04:03:45Z
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Groups of nilpotency class \(2\) of order \(p^4\) as additive groups of local nearrings

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