On One Open Question of the Theory of \(\sigma \)-Properties of a Finite Group

Let \(\sigma =\{\sigma _{i} \mid i\in I\}\) $\sigma =\{{\sigma }_i\mid i\in I\}$ be some partition of the set of all primes and G a finite group. A subgroup A of G is \(\sigma \) $\sigma$ -permutable in G provided G is \(\sigma \) $\sigma$ -full; that is, G has a Hall \(\sigma _{i}\) ${\sigma }_i$ -subgroup for all \(i\in I\) $i\in I$ and A permutes with all such Hall subgroups H of G; that is, \(AH=HA\) $AH=HA$ . Answering the Question 6.4 in Skiba (Probl Phys Math Tech 42(21):89–96, 2014), we get a description of finite \(\sigma \) $\sigma$ -full groups G in which \(\sigma \) $\sigma$ -permutability is a transitive relation.