On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups
Let \(G\) be a finite group and \(\sigma=\{\sigma_i | i\in I\}\) be some partition of the set of all primes. A subgroup \(A\) of \(G\) is said to \(K\)-\(\frak{N}_{\sigma}\)-subnormal in \(G\) if there is a subgroup chain \(A=A_o\leq A_1 \leq \cdots \leq A_n=G\) such that either \(A_{i-1} \unlhd A_i...
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| Дата: | 2024 |
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| Мова: | English |
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Lugansk National Taras Shevchenko University
2024
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-22332024-06-27T08:42:43Z On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups Hussain, Muhammad Tanveer finite group, \(\sigma\)-nilpotent group, \(K\)-lattice saturated formation, Schmidt subgroup 20D10, 20D15, 20D20. Let \(G\) be a finite group and \(\sigma=\{\sigma_i | i\in I\}\) be some partition of the set of all primes. A subgroup \(A\) of \(G\) is said to \(K\)-\(\frak{N}_{\sigma}\)-subnormal in \(G\) if there is a subgroup chain \(A=A_o\leq A_1 \leq \cdots \leq A_n=G\) such that either \(A_{i-1} \unlhd A_i\) or \(A_i/ (A_{i-1})_{A_i}\in \frak{N}_{\sigma}\) for all \(i=1,\ldots, n\), where \(\frak{N}_{\sigma}\) is a hereditary \(K\)-lattice saturated formation of all \(\sigma\)-nilpotent groups. The formation \(\frak{N}_{\sigma}\) is called \(K\)-lattice if in every finite group \(G\) the set \(\mathcal{L}_{K\frak{N}_{\sigma}}(G)\), of all \(K\)-\(\frak{N}_{\sigma}\)-subnormal subgroup of \(G\), is a sublattice of the lattice \(\mathcal{L}(G)\) of all subgroups of \(G\). In this paper we prove that if every Schmidt subgroup of \(G\) is \(K\)-\(\frak{N}_{\sigma}\)-subnormal subgroup of \(G\), then the commutator subgroup \(G'\) of \(G\) belongs to hereditary \(K\)-lattice saturated formation \(\frak{N}_{\sigma}\). Lugansk National Taras Shevchenko University 2024-06-27 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2233 10.12958/adm2233 Algebra and Discrete Mathematics; Vol 37, No 2 (2024) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2233/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2233/1166 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2233/1197 Copyright (c) 2024 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2024-06-27T08:42:43Z |
| collection |
OJS |
| language |
English |
| topic |
finite group \(\sigma\)-nilpotent group \(K\)-lattice saturated formation Schmidt subgroup 20D10 20D15 20D20. |
| spellingShingle |
finite group \(\sigma\)-nilpotent group \(K\)-lattice saturated formation Schmidt subgroup 20D10 20D15 20D20. Hussain, Muhammad Tanveer On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups |
| topic_facet |
finite group \(\sigma\)-nilpotent group \(K\)-lattice saturated formation Schmidt subgroup 20D10 20D15 20D20. |
| format |
Article |
| author |
Hussain, Muhammad Tanveer |
| author_facet |
Hussain, Muhammad Tanveer |
| author_sort |
Hussain, Muhammad Tanveer |
| title |
On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups |
| title_short |
On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups |
| title_full |
On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups |
| title_fullStr |
On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups |
| title_full_unstemmed |
On finite groups with \(K\)-\(\frak{N}_{\sigma}\)-subnormal Schmidt subgroups |
| title_sort |
on finite groups with \(k\)-\(\frak{n}_{\sigma}\)-subnormal schmidt subgroups |
| description |
Let \(G\) be a finite group and \(\sigma=\{\sigma_i | i\in I\}\) be some partition of the set of all primes. A subgroup \(A\) of \(G\) is said to \(K\)-\(\frak{N}_{\sigma}\)-subnormal in \(G\) if there is a subgroup chain \(A=A_o\leq A_1 \leq \cdots \leq A_n=G\) such that either \(A_{i-1} \unlhd A_i\) or \(A_i/ (A_{i-1})_{A_i}\in \frak{N}_{\sigma}\) for all \(i=1,\ldots, n\), where \(\frak{N}_{\sigma}\) is a hereditary \(K\)-lattice saturated formation of all \(\sigma\)-nilpotent groups. The formation \(\frak{N}_{\sigma}\) is called \(K\)-lattice if in every finite group \(G\) the set \(\mathcal{L}_{K\frak{N}_{\sigma}}(G)\), of all \(K\)-\(\frak{N}_{\sigma}\)-subnormal subgroup of \(G\), is a sublattice of the lattice \(\mathcal{L}(G)\) of all subgroups of \(G\). In this paper we prove that if every Schmidt subgroup of \(G\) is \(K\)-\(\frak{N}_{\sigma}\)-subnormal subgroup of \(G\), then the commutator subgroup \(G'\) of \(G\) belongs to hereditary \(K\)-lattice saturated formation \(\frak{N}_{\sigma}\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2024 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2233 |
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2024-06-28T04:03:56Z |
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2024-06-28T04:03:56Z |
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