Finite Groups with \({{{\mathcal {H}}}}\)-Permutable Subgroups

Let \(\sigma =\{\sigma _{i}\ |\ i\in I\}\) be some partition of the set \(\mathbb {P}\) of all primes and G a finite group. A set \({{{\mathcal {H}}}}\) of subgroups of G is said to be a complete Hall \(\sigma \)-set of G if every member \(\ne 1\) of \({{{\mathcal {H}}}}\) is a Hall \(\sigma _{i}\)-subgroup of G for some \(i\in I\) and \({{\mathcal {H}}}\) contains exactly one Hall \(\sigma _{i}\)-subgroup of G for every i such that \(\sigma _{i}\cap \pi (G)\ne \emptyset \). A subgroup A of G is said to be \({{\mathcal {H}}}\)-permutable if A permutes with all members of the complete Hall \(\sigma \)-set \({{{\mathcal {H}}}}\) of G. In this paper, we study the structure of G under the assuming that some subgroups of G are \({{\mathcal {H}}}\)-permutable.