Finite groups with semi-subnormal Schmidt subgroups
A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup \(A\) of a group \(G\) is semi-normal in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=AB\) and \(AB_1\) is a proper subgroup of \(G\) for every proper subgroup \(B_1\) of \(B\). If \(A\)...
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| Date: | 2020 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2020
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1376 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543379104464896 |
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| author | Kniahina, V. N. Monakhov, V. S. |
| author_facet | Kniahina, V. N. Monakhov, V. S. |
| author_sort | Kniahina, V. N. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2020-05-14T18:27:22Z |
| description | A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup \(A\) of a group \(G\) is semi-normal in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=AB\) and \(AB_1\) is a proper subgroup of \(G\) for every proper subgroup \(B_1\) of \(B\). If \(A\) is either subnormal in \(G\) or is semi-normal in \(G\), then \(A\) is called a semi-subnormal subgroup of \(G\). In this paper, we establish that a group \(G\) with semi-subnormal Schmidt \(\{2,3\}\)-subgroups is \(3\)-soluble. Moreover, if all 5-closed Schmidt \(\{2,5 \}\)-subgroups are semi-subnormal in \(G\), then \(G\) is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent. |
| first_indexed | 2025-12-02T15:38:49Z |
| format | Article |
| id | admjournalluguniveduua-article-1376 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:38:49Z |
| publishDate | 2020 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-13762020-05-14T18:27:22Z Finite groups with semi-subnormal Schmidt subgroups Kniahina, V. N. Monakhov, V. S. finite soluble group, Schmidt subgroup, semi-normal subgroup, subnormal subgroup 20E28, 20E32, 20E34 A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup \(A\) of a group \(G\) is semi-normal in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=AB\) and \(AB_1\) is a proper subgroup of \(G\) for every proper subgroup \(B_1\) of \(B\). If \(A\) is either subnormal in \(G\) or is semi-normal in \(G\), then \(A\) is called a semi-subnormal subgroup of \(G\). In this paper, we establish that a group \(G\) with semi-subnormal Schmidt \(\{2,3\}\)-subgroups is \(3\)-soluble. Moreover, if all 5-closed Schmidt \(\{2,5 \}\)-subgroups are semi-subnormal in \(G\), then \(G\) is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent. 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/1376 10.12958/adm1376 Algebra and Discrete Mathematics; Vol 29, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1376/pdf Copyright (c) 2020 Algebra and Discrete Mathematics |
| spellingShingle | finite soluble group Schmidt subgroup semi-normal subgroup subnormal subgroup 20E28 20E32 20E34 Kniahina, V. N. Monakhov, V. S. Finite groups with semi-subnormal Schmidt subgroups |
| title | Finite groups with semi-subnormal Schmidt subgroups |
| title_full | Finite groups with semi-subnormal Schmidt subgroups |
| title_fullStr | Finite groups with semi-subnormal Schmidt subgroups |
| title_full_unstemmed | Finite groups with semi-subnormal Schmidt subgroups |
| title_short | Finite groups with semi-subnormal Schmidt subgroups |
| title_sort | finite groups with semi-subnormal schmidt subgroups |
| topic | finite soluble group Schmidt subgroup semi-normal subgroup subnormal subgroup 20E28 20E32 20E34 |
| topic_facet | finite soluble group Schmidt subgroup semi-normal subgroup subnormal subgroup 20E28 20E32 20E34 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1376 |
| work_keys_str_mv | AT kniahinavn finitegroupswithsemisubnormalschmidtsubgroups AT monakhovvs finitegroupswithsemisubnormalschmidtsubgroups |