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The Influence of Weakly \({\sigma }\)-Permutably Embedded Subgroups on the Structure of Finite Groups

  • Muhammad Tanveer Hussain
  • Chenchen Cao
  • Chi ZhangEmail author
Original Paper
  • 22 Downloads

Abstract

Let \(\sigma =\{{\sigma _i|i\in I}\}\) be some partition of the set of all primes \({\mathbb {P}}\), G a finite group and \(\sigma (G)=\{{\sigma _i|\sigma _i \cap \pi (G) \ne \emptyset }\}\). A set \({\mathcal {H}} \) of subgroups of G is said to be a complete Hall\(\sigma \)-set of G if every non-identity member of \({\mathcal {H}}\) is a Hall \(\sigma _i\)-subgroup of G and \({\mathcal {H}}\) contains exactly one Hall \(\sigma _i\)-subgroup of G for every \(\sigma _i\in \sigma (G)\). G is said to be \(\sigma \)-full if G possesses a complete Hall \(\sigma \)-set. A subgroup H of G is said to be \(\sigma \)-permutable in G provided there is a complete Hall \(\sigma \)-set \({\mathcal {H}}\) of G such that \(HA^x=A^xH\) for all \(A\in {\mathcal {H}}\) and all \(x\in G\); \(\sigma \)-permutably embedded in G if H is \(\sigma \)-full and for every \(\sigma _i \in \sigma (H)\), every Hall \(\sigma _i\)-subgroup of H is also a Hall \(\sigma _i\)-subgroup of some \(\sigma \)-permutable subgroup of G. We call that a subgroup H of G is weakly\({\sigma }\)-permutably embedded in G if there exists a \(\sigma \)-subnormal subgroup T of G such that \(G=HT\) and \(H\cap T\le H_{\sigma eG}\), where \(H_{\sigma eG}\) is the subgroup of H generated by all those subgroups of H which are \(\sigma \)-permutably embedded in G. In this paper, we study the structure of G under the condition that some given subgroups of G are weakly \({\sigma }\)-permutably embedded in G. Some known results are generalized.

Keywords

Finite groups \(\sigma \)-Permutable subgroup \(\sigma \)-Permutably embedded subgroup Weakly \({\sigma }\)-permutably embedded subgroup Supersoluble group 

Mathematics Subject Classification

20D10 20D15 20D20 20D35 

Notes

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  • Muhammad Tanveer Hussain
    • 1
  • Chenchen Cao
    • 2
  • Chi Zhang
    • 3
    Email author
  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.School of Mathematics and statisticsNingbo UniversityNingboPeople’s Republic of China
  3. 3.Department of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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