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Siberian Mathematical Journal

, Volume 60, Issue 4, pp 720–726 | Cite as

On Strongly Π-Permutable Subgroups of a Finite Group

  • B. Hu
  • J. HuangEmail author
  • A. N. Skiba
Article
  • 7 Downloads

Abstract

Let σ = {σi | iI} be some partition of the set of all primes ℙ,let ∅ ≠ Π ⊆ σ, and let G be a finite group. A set of subgroups of G is said to be a complete Hall Π-set of G if each member ≠ 1 of is a Hall σi-subgroup of G for some σi ∈ Π and has exactly one Hall σi-subgroup of G for every σi ∈ Π such that σiπ(G) ≠ ∅. A subgroup A of G is called (i) Π-permutable in G if G has a complete Hall Π-set such that AHx = HxA for all H and xG; (ii) σ-subnormal in G if there is a subgroup chain A = A0A1 ≤ ⋯ ≤ At = G such that either Ai−1Ai or Ai/(Ai−1)Ai is a σk-group for some k for all i = 1,…,t; and (iii) strongly Π-permutable if A is Π-permutable and σ-subnormal in G. We study the strongly Π-permutable subgroups of G. In particular, we give characterizations of these subgroups and prove that the set of all strongly Π-permutable subgroups of G forms a sublattice of the lattice of all subgroups of G.

Keywords

finite group subgroup lattice σ-subnormal subgroup strongly Π-permutable subgroup 

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Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouP. R. China
  2. 2.Francisk Skorina Gomel State UniversityGomelBelarus

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