On \(S\)-quasinormally embedded subgroups of finite groups
Let \(G\) be a finite group. A subgroup \(A\) is called: 1) \(S\)-quasinormal in \(G\) if \(A\) is permutable with all Sylow subgroups in \(G\) 2) \(S\)-quasinormally embedded in \(G\) if every Sylow subgroup of \(A\) is a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). Let \(B_{seG}\)...
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| Date: | 2018 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/689 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | Let \(G\) be a finite group. A subgroup \(A\) is called: 1) \(S\)-quasinormal in \(G\) if \(A\) is permutable with all Sylow subgroups in \(G\) 2) \(S\)-quasinormally embedded in \(G\) if every Sylow subgroup of \(A\) is a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). Let \(B_{seG}\) be the subgroup generated by all the subgroups of \(B\) which are \(S\)-quasinormally embedded in \(G\). A subgroup \(B\) is called \(SE\)-supplemented in \(G\) if there exists a subgroup \(T\) such that \(G=BT\) and \(B\cap T\le B_{seG}\). The main result of the paper is the following.Theorem. Let \(H\) be a normal subgroup in \(G\), and \(p\) a prime divisor of \(|H|\) such that \((p-1,|H|)=1\). Let \(P\) be a Sylow \(p\)-subgroup in \(H\). Assume that all maximal subgroups in \(P\) are \(SE\)-supplemented in \(G\). Then \(H\) is \(p\)-nilpotent and all its \(G\)-chief \(p\)-factors are cyclic. |
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