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Title: | A note on Hall S-permutably embedded subgroups of finite groups |
Authors: | Sinitsa, D.A. |
Keywords: | S-permutable subgroup Hall S-permutably embed- ded subgroup Sylow subgroup supersoluble group maximal subgroup |
Issue Date: | 2017 |
Publisher: | ДЗ "ЛНУ імені Тараса Шевченка" |
Series/Report no.: | математичні науки; |
Abstract: | 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. |
Description: | Sinitsa D.A. A note on Hall S-permutably embedded subgroups of finite groups / D.A.Sinitsa // Algebra and Discrete Mathematics. - 2017. - Vol. 23. - Number 2. - Рp.305- 311 |
URI: | http://hdl.handle.net/123456789/4675 |
Appears in Collections: | Algebra and Discrete Mathematics. - № 2 (23). - 2017 |
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