On -permutable subgroups in finite groups

UDC 512.542Let be some partition of the set of all primes and let be a nonempty subset of the set   A set of subgroups of a finite group is said to be a \emph{complete Hall -set} of if every member of is a Hall -subgroup of for some and contains exactly one Hall -subgroup of for every such that  A subgroup of is called (i) {- permutable } if for and ; (ii) {- permutable in } if is -permutable for some complete Hall -set of  We study the influence of -permutable subgroups on the structure of  In particular, we prove that if and where and are -permutable -separable (respectively, -closed) subgroups of then is also -separable (respectively, -closed).  Some known results are generalized.