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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| Дата: | 2020 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2020
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1479 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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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 |
| baseUrl_str |
|
| datestamp_date |
2020-05-14T18:27:22Z |
| 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 |
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AT stockaa setsofprimepowerordergeneratorsoffinitegroups |
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2025-07-17T10:35:12Z |
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2025-07-17T10:35:12Z |
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