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Sets of prime power order generators of finite groups

Sets of prime power order generators of finite groups

A subset \(X\) of prime power order elements of a finite group \(G\) is called pp-independent if there is no proper subset \(Y\) of \(X\) such that \(\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). A group \(G\) has property \(\mathcal{B}...

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Main Author: Stocka, A.
Format: Article
Language:English
Published: Lugansk National Taras Shevchenko University 2020
Subjects:
finite groups
independent sets
minimal generating sets
Burnside basis theorem
Primary 20D10; Secondary 20F05
Online Access:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1479
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spelling oai:ojs.admjournal.luguniv.edu.ua:article-14792020-05-14T18:27:22Z Sets of prime power order generators of finite groups Stocka, A. finite groups, independent sets, minimal generating sets, Burnside basis theorem Primary 20D10; Secondary 20F05 A subset \(X\) of prime power order elements of a finite group \(G\) is called pp-independent if there is no proper subset \(Y\) of \(X\) such that \(\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). A group \(G\) has property \(\mathcal{B}_{pp}\) if all pp-independent generating sets of \(G\) have the same size. \(G\) has the pp-basis exchange property if for any pp-independent generating sets \(B_1, B_2\) of \(G\) and \(x\in B_1\) there exists \(y\in B_2\) such that \((B_1\setminus \{x\})\cup \{y\}\) is a pp-independent generating set of \(G\). In this paper we describe all finite solvable groups with property \(\mathcal{B}_{pp}\) and all finite solvable groups with the pp-basis exchange property. Lugansk National Taras Shevchenko University This article has received financial support from the Polish Ministry of Science and Higher Education under subsidy for maintaining the research potential of the Faculty of Mathematics and Informatics, University of Bia{\l}ystok 2020-05-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1479 10.12958/adm1479 Algebra and Discrete Mathematics; Vol 29, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1479/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1479/611 Copyright (c) 2020 Algebra and Discrete Mathematics
institution Algebra and Discrete Mathematics
collection OJS
language English
topic finite groups
independent sets
minimal generating sets
Burnside basis theorem
Primary 20D10; Secondary 20F05
spellingShingle finite groups
independent sets
minimal generating sets
Burnside basis theorem
Primary 20D10; Secondary 20F05
Stocka, A.
Sets of prime power order generators of finite groups
topic_facet finite groups
independent sets
minimal generating sets
Burnside basis theorem
Primary 20D10; Secondary 20F05
format Article
author Stocka, A.
author_facet Stocka, A.
author_sort Stocka, A.
title Sets of prime power order generators of finite groups
title_short Sets of prime power order generators of finite groups
title_full Sets of prime power order generators of finite groups
title_fullStr Sets of prime power order generators of finite groups
title_full_unstemmed Sets of prime power order generators of finite groups
title_sort sets of prime power order generators of finite groups
description A subset \(X\) of prime power order elements of a finite group \(G\) is called pp-independent if there is no proper subset \(Y\) of \(X\) such that \(\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). A group \(G\) has property \(\mathcal{B}_{pp}\) if all pp-independent generating sets of \(G\) have the same size. \(G\) has the pp-basis exchange property if for any pp-independent generating sets \(B_1, B_2\) of \(G\) and \(x\in B_1\) there exists \(y\in B_2\) such that \((B_1\setminus \{x\})\cup \{y\}\) is a pp-independent generating set of \(G\). In this paper we describe all finite solvable groups with property \(\mathcal{B}_{pp}\) and all finite solvable groups with the pp-basis exchange property.
publisher Lugansk National Taras Shevchenko University
publishDate 2020
url https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1479
work_keys_str_mv AT stockaa setsofprimepowerordergeneratorsoffinitegroups
first_indexed 2024-04-12T06:25:38Z
last_indexed 2024-04-12T06:25:38Z
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