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Siberian Mathematical Journal

, Volume 60, Issue 3, pp 389–397 | Cite as

On σ-Embedded and σ-n-Embedded Subgroups of Finite Groups

  • V. AmjidEmail author
  • W. GuoEmail author
  • B. LiEmail author
Article
  • 19 Downloads

Abstract

Let G be a finite group, and let σ = {σi | iI} be a partition of the set of all primes ℙ and σ(G) = {σi | σiπ(G) ≠ ∅}. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σi-subgroup of G and ℋ has exactly one Hall σi-subgroup of G for every σiσ (G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HAx = AxH for all A ∈ ℋ and xG. A subgroup H of G is said to be σ-n-embedded in G if there exists a normal subgroup T of G such that HT = HG and HTHσG, where HσG is the subgroup of H generated by all those subgroups of H that are σ-permutable in G. A subgroup H of G is said to be σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = HσG and HTHσG, where HσG is the intersection of all σ-permutable subgroups of G containing H. We study the structure of finite groups under the condition that some given subgroups of G are σ-embedded and σ-n-embedded. In particular, we give the conditions for a normal subgroup of G to be hypercyclically embedded.

Keywords

finite group σ-embedded subgroup σ-n-embedded subgroup supersoluble hypercyclically embedded 

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Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiP. R. China
  2. 2.College of Applied MathematicsChengdu University of Information TechnologyChengduP. R. China

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