The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs
Base (minimal generating set) of the Sylow 2-subgroup of \(S_{2^n}\) is called diagonal if every element of this set acts non-trivially only on one coordinate, and different elements act on different coordinates. The Sylow 2-subgroup \(P_n(2)\) of \(S_{2^n}\) acts by conjugation on the set of all b...
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| Datum: | 2016 |
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Lugansk National Taras Shevchenko University
2016
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/233 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543388070838272 |
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| author | Pawlik, Bartłomiej Tadeusz |
| author_facet | Pawlik, Bartłomiej Tadeusz |
| author_sort | Pawlik, Bartłomiej Tadeusz |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2016-07-12T10:09:40Z |
| description | Base (minimal generating set) of the Sylow 2-subgroup of \(S_{2^n}\) is called diagonal if every element of this set acts non-trivially only on one coordinate, and different elements act on different coordinates. The Sylow 2-subgroup \(P_n(2)\) of \(S_{2^n}\) acts by conjugation on the set of all bases. In presented paper the~stabilizer of the set of all diagonal bases in \(S_n(2)\) is characterized and the orbits of the action are determined. It is shown that every orbit contains exactly \(2^{n-1}\) diagonal bases and \(2^{2^n-2n}\) bases at all. Recursive construction of Cayley graphs of \(P_n(2)\) on diagonal bases (\(n\geq2\)) is proposed. |
| first_indexed | 2025-12-02T15:42:39Z |
| format | Article |
| id | admjournalluguniveduua-article-233 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:42:39Z |
| publishDate | 2016 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-2332016-07-12T10:09:40Z The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs Pawlik, Bartłomiej Tadeusz Sylow \(p\)-subgroup, group base, wreath product of groups, Cayley graphs 20B35, 20D20, 20E22, 05C25 Base (minimal generating set) of the Sylow 2-subgroup of \(S_{2^n}\) is called diagonal if every element of this set acts non-trivially only on one coordinate, and different elements act on different coordinates. The Sylow 2-subgroup \(P_n(2)\) of \(S_{2^n}\) acts by conjugation on the set of all bases. In presented paper the~stabilizer of the set of all diagonal bases in \(S_n(2)\) is characterized and the orbits of the action are determined. It is shown that every orbit contains exactly \(2^{n-1}\) diagonal bases and \(2^{2^n-2n}\) bases at all. Recursive construction of Cayley graphs of \(P_n(2)\) on diagonal bases (\(n\geq2\)) is proposed. Lugansk National Taras Shevchenko University 2016-07-12 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/233 Algebra and Discrete Mathematics; Vol 21, No 2 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/233/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/233/92 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/233/93 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/233/94 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/233/95 Copyright (c) 2016 Algebra and Discrete Mathematics |
| spellingShingle | Sylow \(p\)-subgroup group base wreath product of groups Cayley graphs 20B35 20D20 20E22 05C25 Pawlik, Bartłomiej Tadeusz The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs |
| title | The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs |
| title_full | The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs |
| title_fullStr | The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs |
| title_full_unstemmed | The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs |
| title_short | The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs |
| title_sort | action of sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their cayley graphs |
| topic | Sylow \(p\)-subgroup group base wreath product of groups Cayley graphs 20B35 20D20 20E22 05C25 |
| topic_facet | Sylow \(p\)-subgroup group base wreath product of groups Cayley graphs 20B35 20D20 20E22 05C25 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/233 |
| work_keys_str_mv | AT pawlikbartłomiejtadeusz theactionofsylow2subgroupsofsymmetricgroupsonthesetofbasesandtheproblemofisomorphismoftheircayleygraphs AT pawlikbartłomiejtadeusz actionofsylow2subgroupsofsymmetricgroupsonthesetofbasesandtheproblemofisomorphismoftheircayleygraphs |