On Strongly Π-Permutable Subgroups of a Finite Group

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.