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Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 11–24 | Cite as

On σ-Permutably Embedded Subgroups of Finite Groups

  • Chenchen CaoEmail author
  • Li Zhang
  • Wenbin Guo
Article
  • 44 Downloads

Abstract

Let σ = {σi: iI} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi: σi ∩ π(G)≠Ø}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σi-subgroup of G and H contains exactly one Hall σi-subgroup of G for every σiσ(G). G is said to be σ-full if G possesses a complete Hall σ-set. A subgroup H of G is σ-permutable in G if G possesses a complete Hall σ-set H such that HAx= AxH for all AH and all xG. A subgroup H of G is σ-permutably embedded in G if H is σ-full and for every σiσ(H), every Hall σi-subgroup of H is also a Hall σi-subgroup of some σ-permutable subgroup of G.

By using the σ-permutably embedded subgroups, we establish some new criteria for a group G to be soluble and supersoluble, and also give the conditions under which a normal subgroup of G is hypercyclically embedded. Some known results are generalized.

Keywords

finite group σ-subnormal subgroup σ-permutably embedded subgroup σ-soluble group supersoluble group 

MSC 2010

20D10 20D20 20D35 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefei, AnhuiP.R. China
  2. 2.School of Mathmatics and PhysicsAnhui Jianzhu UniversityHefei, AuhuiP.R. China

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