On the lattice of the σ-permutable subgroups of a finite group

Abstract

Let σ = {σi|i ∈ I} be some partition of the set of all primes P, G a finite group and σ(G) = {σi|σi ∩ π(G) 6= ∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member 6= 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup A of G is said to be σ-permutable in G if G possesses a complete Hall σ-set and A permutes with each Hall σi-subgroup H of G, that is, AH = HA for all i ∈ I. We characterize finite groups with distributive lattice of the σ-permutable subgroups.