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.
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Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 4, pp. 922–932.
The authors were supported by the NNSF of China (Grant 11401264) and a TAPP of Jiangsu Higher Education Institutions (Grant PPZY 2015A013).
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Hu, B., Huang, J. & Skiba, A.N. On Strongly Π-Permutable Subgroups of a Finite Group. Sib Math J 60, 720–726 (2019). https://doi.org/10.1134/S0037446619040177
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DOI: https://doi.org/10.1134/S0037446619040177