$A new characterization of finite $\sigma$-soluble $P\sigma T$-groups$

A new characterization of finite $\sigma$-soluble $P\sigma T$-groups

Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A set ${\mathcal H}$ of subgroups of \(G...

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Дата:	2020
Автор:	Adarchenko, N. M.
Формат:	Стаття
Мова:	English
Опубліковано:	Lugansk National Taras Shevchenko University 2020
Теми:	finite group $\sigma$-permutable subgroup $P\sigma T$-group $\sigma$-soluble group $\sigma$-nilpotent group 20D10 20D15 20D30
Онлайн доступ:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530
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

id	oai:ojs.admjournal.luguniv.edu.ua:article-1530
record_format	ojs
spelling	oai:ojs.admjournal.luguniv.edu.ua:article-15302020-05-14T18:27:22Z A new characterization of finite $\sigma$-soluble $P\sigma T$-groups Adarchenko, N. M. finite group, $\sigma$-permutable subgroup, $P\sigma T$-group, $\sigma$-soluble group, $\sigma$-nilpotent group 20D10, 20D15, 20D30 Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A set ${\mathcal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\mathcal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\mathcal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$ such that $\sigma_{i}\cap \pi (G)\ne \varnothing$. A subgroup $A$ of $G$ is said to be ${\sigma}$-quasinormal or ${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\mathcal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \mathcal H$. We obtain a new characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$. Lugansk National Taras Shevchenko University 2020-05-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530 10.12958/adm1530 Algebra and Discrete Mathematics; Vol 29, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1530/656 Copyright (c) 2020 Algebra and Discrete Mathematics
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2020-05-14T18:27:22Z
collection	OJS
language	English
topic	finite group $\sigma$-permutable subgroup $P\sigma T$-group $\sigma$-soluble group $\sigma$-nilpotent group 20D10 20D15 20D30
spellingShingle	finite group $\sigma$-permutable subgroup $P\sigma T$-group $\sigma$-soluble group $\sigma$-nilpotent group 20D10 20D15 20D30 Adarchenko, N. M. A new characterization of finite $\sigma$-soluble $P\sigma T$-groups
topic_facet	finite group $\sigma$-permutable subgroup $P\sigma T$-group $\sigma$-soluble group $\sigma$-nilpotent group 20D10 20D15 20D30
format	Article
author	Adarchenko, N. M.
author_facet	Adarchenko, N. M.
author_sort	Adarchenko, N. M.
title	A new characterization of finite $\sigma$-soluble $P\sigma T$-groups
title_short	A new characterization of finite $\sigma$-soluble $P\sigma T$-groups
title_full	A new characterization of finite $\sigma$-soluble $P\sigma T$-groups
title_fullStr	A new characterization of finite $\sigma$-soluble $P\sigma T$-groups
title_full_unstemmed	A new characterization of finite $\sigma$-soluble $P\sigma T$-groups
title_sort	new characterization of finite $\sigma$-soluble $p\sigma t$-groups
description	Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A set ${\mathcal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\mathcal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\mathcal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$ such that $\sigma_{i}\cap \pi (G)\ne \varnothing$. A subgroup $A$ of $G$ is said to be ${\sigma}$-quasinormal or ${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\mathcal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \mathcal H$. We obtain a new characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$.
publisher	Lugansk National Taras Shevchenko University
publishDate	2020
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530
work_keys_str_mv	AT adarchenkonm anewcharacterizationoffinitesigmasolublepsigmatgroups AT adarchenkonm newcharacterizationoffinitesigmasolublepsigmatgroups
first_indexed	2025-07-17T10:30:59Z
last_indexed	2025-07-17T10:30:59Z
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A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups

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