On σ-subnormal and σ-permutable subgroups of finite groups

Let σ={σi|i∈I}σ={σi|i∈I} be some partition of the set PP of all primes, that is, P=∪i∈IσiP=∪i∈Iσi and σi∩σj=∅σi∩σj=∅ for all i≠ji≠j. Let G be a finite group. We say that G is: σ-primary if G   is a σiσi-group for some i∈Ii∈I; σ-soluble if every chief factor of G is σ  -primary. We say that a set H={H1,…,Ht}H={H1,…,Ht} of Hall subgroups of G  , where HiHi is σ  -primary (i=1,…,ti=1,…,t), is a complete Hall set of type σ of G   if (|Hi|,|Hj|)=1(|Hi|,|Hj|)=1 for all i≠ji≠j and π(G)=π(H1)∪⋯∪π(Ht)π(G)=π(H1)∪⋯∪π(Ht). We say that a subgroup A of G is: σ-subnormal in G   if there is a subgroup chain A=A0≤A1≤⋯≤An=GA=A0≤A1≤⋯≤An=G such that either Ai−1Ai−1 is normal in AiAi or Ai/(Ai−1)AiAi/(Ai−1)Ai is σ  -primary for all i=1,…,ti=1,…,t; σ-permutable in G if G   has a complete Hall set HH of type σ such that A   is HGHG-permutable in G  , that is, AHx=HxAAHx=HxA for all x∈Gx∈G and all H∈HH∈H. We study the relationship between the σ-subnormal and σ-permutable subgroups of G. In particular, we prove that every σ-permutable subgroup of G is σ-subnormal, and we classify finite σ-soluble groups in which every σ-subnormal subgroup is σ-permutable.