On Strongly Π-Permutable Subgroups of a Finite Group
- Authors: Hu B.1, Huang J.1, Skiba A.N.2
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Affiliations:
- School of Mathematics and Statistics
- Francisk Skorina Gomel State University
- Issue: Vol 60, No 4 (2019)
- Pages: 720-726
- Section: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/172568
- DOI: https://doi.org/10.1134/S0037446619040177
- ID: 172568
Cite item
Abstract
Let σ = {σi | i ∈ I} be some partition of the set of all primes ℙ,let ∅ ≠ Π ⊆ σ, and let G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall Π-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some σi ∈ Π and ℋ has exactly one Hall σi-subgroup of G for every σi ∈ Π such that σi ∩ π(G) ≠ ∅. A subgroup A of G is called (i) Π-permutable in G if G has a complete Hall Π-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G; (ii) σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ ⋯ ≤ At = G such that either Ai−1 ≤ Ai 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.
About the authors
B. Hu
School of Mathematics and Statistics
Email: jhh320@126.com
China, Xuzhou
J. Huang
School of Mathematics and Statistics
Author for correspondence.
Email: jhh320@126.com
China, Xuzhou
A. N. Skiba
Francisk Skorina Gomel State University
Email: jhh320@126.com
Belarus, Gomel