On 𝜎-permutable subgroups of 𝜎-soluble finite groups

Let σ = { σ i ∣ i ∈ I } \sigma=\{\sigma_{i}\mid i\in I\} be some partition of the set of all primes and 𝐺 a finite group. Then 𝐺 is said to be 𝜎-full if 𝐺 has a Hall σ i \sigma_{i} -subgroup for all i ∈ I i\in I and 𝜎-primary if 𝐺 is a σ i \sigma_{i} -group for some 𝑖. In addition, 𝐺 is 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary and 𝜎-nilpotent if 𝐺 is a direct product of 𝜎-primary groups. We write G N σ G^{\mathfrak{N}_{\sigma}} for the 𝜎-nilpotent residual of 𝐺, which is the intersection of all normal subgroups 𝑁 of 𝐺 with 𝜎-nilpotent G / N G/N . A subgroup 𝐴 of 𝐺 is said to be 𝜎-permutable in 𝐺 provided 𝐺 is 𝜎-full and 𝐴 permutes with all Hall σ i \sigma_{i} -subgroups 𝐻 of 𝐺 (that is, A ⁢ H = H ⁢ A AH=HA ) for all 𝑖. And 𝐴 is 𝜎-subnormal in 𝐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G A=A_{0}\leq A_{1}\leq\cdots\leq A_{n}=G such that either A i − 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i − 1 ) A i A_{i}/(A_{i-1})_{A_{i}} is 𝜎-primary for all i = 1 , … , n i=1,\ldots,n . We prove that if 𝐺 is a 𝜎-soluble group, then 𝜎-permutability is a transitive relation in 𝐺 if and only if G N σ ∩ A G = G N σ ∩ A G G^{\mathfrak{N}_{\sigma}}\cap A^{G}=G^{\mathfrak{N}_{\sigma}}\cap A_{G} for every 𝜎-subnormal subgroup 𝐴 of 𝐺.