A note on Hall S-permutably embedded subgroups of finite groups
Let G be a finite group. Recall that a subgroup A of G is said to permute with a subgroup B if AB=BA. A subgroup A of G is said to be S-quasinormal or S-permutable in G if A permutes with all Sylow subgroups of G. Recall also that HsG is the S-permutable closure of H in G, that is, the intersect...
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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2017 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2017
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/156024 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A note on Hall S-permutably embedded subgroups of finite groups / D. Sinitsa // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 305-311. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862697492362035200 |
|---|---|
| author | Sinitsa, D. |
| author_facet | Sinitsa, D. |
| citation_txt | A note on Hall S-permutably embedded subgroups of finite groups / D. Sinitsa // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 305-311. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Let G be a finite group. Recall that a subgroup A of G is said to permute with a subgroup B if AB=BA. A subgroup A of G is said to be S-quasinormal or S-permutable in G if A permutes with all Sylow subgroups of G. Recall also that HsG is the S-permutable closure of H in G, that is, the intersection of all such S-permutable subgroups of G which contain H. We say that H is Hall S-permutably embedded in G if H is a Hall subgroup of the S-permutable closure HsG of H in G. We prove that the following conditions are equivalent: (1) every subgroup of G is Hall S-permutably embedded in G; (2) the nilpotent residual GN of G is a Hall cyclic of square-free order subgroup of G; (3) G=D⋊M is a split extension of a cyclic subgroup D of square-free order by a nilpotent group M, where M and D are both Hall subgroups of G.
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| first_indexed | 2025-12-07T16:29:58Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-156024 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T16:29:58Z |
| publishDate | 2017 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Sinitsa, D. 2019-06-17T18:59:06Z 2019-06-17T18:59:06Z 2017 A note on Hall S-permutably embedded subgroups of finite groups / D. Sinitsa // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 305-311. — Бібліогр.: 9 назв. — англ. 1726-3255 2010 MSC:20D10, 20D15, 20D30. https://nasplib.isofts.kiev.ua/handle/123456789/156024 Let G be a finite group. Recall that a subgroup A of G is said to permute with a subgroup B if AB=BA. A subgroup A of G is said to be S-quasinormal or S-permutable in G if A permutes with all Sylow subgroups of G. Recall also that HsG is the S-permutable closure of H in G, that is, the intersection of all such S-permutable subgroups of G which contain H. We say that H is Hall S-permutably embedded in G if H is a Hall subgroup of the S-permutable closure HsG of H in G. We prove that the following conditions are equivalent: (1) every subgroup of G is Hall S-permutably embedded in G; (2) the nilpotent residual GN of G is a Hall cyclic of square-free order subgroup of G; (3) G=D⋊M is a split extension of a cyclic subgroup D of square-free order by a nilpotent group M, where M and D are both Hall subgroups of G. The author is very grateful for the helpful suggestions and remarks of the referee. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics A note on Hall S-permutably embedded subgroups of finite groups Article published earlier |
| spellingShingle | A note on Hall S-permutably embedded subgroups of finite groups Sinitsa, D. |
| title | A note on Hall S-permutably embedded subgroups of finite groups |
| title_full | A note on Hall S-permutably embedded subgroups of finite groups |
| title_fullStr | A note on Hall S-permutably embedded subgroups of finite groups |
| title_full_unstemmed | A note on Hall S-permutably embedded subgroups of finite groups |
| title_short | A note on Hall S-permutably embedded subgroups of finite groups |
| title_sort | note on hall s-permutably embedded subgroups of finite groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156024 |
| work_keys_str_mv | AT sinitsad anoteonhallspermutablyembeddedsubgroupsoffinitegroups AT sinitsad noteonhallspermutablyembeddedsubgroupsoffinitegroups |