Orbit isomorphic skeleton groups
Recent development in the classification of \(p\)-groups often concentrate on the coclass graph \(\mathcal{G}(p,r)\) associated with the finite \(p\)-groups coclass \(r\), specially on periodicity results on these graphs. In particular, the structure of the subgraph induced by `skeleton groups'...
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| Date: | 2023 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2023
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1886 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543424996442112 |
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| author | Saha, S. |
| author_facet | Saha, S. |
| author_sort | Saha, S. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2023-10-30T03:23:03Z |
| description | Recent development in the classification of \(p\)-groups often concentrate on the coclass graph \(\mathcal{G}(p,r)\) associated with the finite \(p\)-groups coclass \(r\), specially on periodicity results on these graphs. In particular, the structure of the subgraph induced by `skeleton groups' is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in \(\mathcal{G}(p,1)\). We find a family of skeleton groups in \(\mathcal{G}(7,1)\) whose \(6\)-step parent is not a periodic parent. This shows that the periodicity results available in the current literature for primes \(p\equiv 5\bmod 6\) do not hold for the primes \(p\equiv 1\bmod 6\). We also improve a known periodicity result in a special case of skeleton groups. |
| first_indexed | 2026-02-08T07:59:23Z |
| format | Article |
| id | admjournalluguniveduua-article-1886 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:59:23Z |
| publishDate | 2023 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-18862023-10-30T03:23:03Z Orbit isomorphic skeleton groups Saha, S. finite groups, \(p\)-groups, coclass 20D15, 20D99 Recent development in the classification of \(p\)-groups often concentrate on the coclass graph \(\mathcal{G}(p,r)\) associated with the finite \(p\)-groups coclass \(r\), specially on periodicity results on these graphs. In particular, the structure of the subgraph induced by `skeleton groups' is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in \(\mathcal{G}(p,1)\). We find a family of skeleton groups in \(\mathcal{G}(7,1)\) whose \(6\)-step parent is not a periodic parent. This shows that the periodicity results available in the current literature for primes \(p\equiv 5\bmod 6\) do not hold for the primes \(p\equiv 1\bmod 6\). We also improve a known periodicity result in a special case of skeleton groups. Lugansk National Taras Shevchenko University Monash University RTP Scholarship, Australia 2023-10-12 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1886 10.12958/adm1886 Algebra and Discrete Mathematics; Vol 35, No 2 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1886/pdf Copyright (c) 2023 Algebra and Discrete Mathematics |
| spellingShingle | finite groups \(p\)-groups coclass 20D15 20D99 Saha, S. Orbit isomorphic skeleton groups |
| title | Orbit isomorphic skeleton groups |
| title_full | Orbit isomorphic skeleton groups |
| title_fullStr | Orbit isomorphic skeleton groups |
| title_full_unstemmed | Orbit isomorphic skeleton groups |
| title_short | Orbit isomorphic skeleton groups |
| title_sort | orbit isomorphic skeleton groups |
| topic | finite groups \(p\)-groups coclass 20D15 20D99 |
| topic_facet | finite groups \(p\)-groups coclass 20D15 20D99 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1886 |
| work_keys_str_mv | AT sahas orbitisomorphicskeletongroups |