On the separability of subgroups of nilpotent groups by root classes of groups

Abstract

Abstract Suppose that 𝒞 is a class of groups consisting only of periodic groups and P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} is the set of prime numbers that do not divide the order of any element of a 𝒞-group. It is easy to see that if a subgroup 𝑌 of a group 𝑋 is 𝒞-separable in this group, then it is P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated in 𝑋. Let us say that 𝑋 has the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} if all of its P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated subgroups are 𝒞-separable. We find a condition that is sufficient for a nilpotent group 𝑁 to have the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} provided 𝒞 is a root class. We also prove that if 𝑁 is torsion-free, then the indicated condition is necessary for this group to have C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} .