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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| Datum: | 2017 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2017
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| Schriftenreihe: | Algebra and Discrete Mathematics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/156024 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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| Zusammenfassung: | 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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