On Strongly Π-Permutable Subgroups of a Finite Group

Abstract

Let σ = {σ_i | 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 σ_i-subgroup of G for some σ_i ∈ Π and ℋ has exactly one Hall σ_i-subgroup of G for every σ_i ∈ Π such that σ_i ∩ 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 AH^x = H^xA for all H ∈ ℋ and x ∈ G; (ii) σ-subnormal in G if there is a subgroup chain A = A₀ ≤ A₁ ≤ ⋯ ≤ A_t = G such that either A_i−1 ≤ A_i or A_i/(A_i−1)A_i is a σ_k-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.