On conjugate separability of nilpotent subgroups

Let 𝐺 be a group and 𝑘 a positive integer. We say that 𝐺 is conjugate separable abelian (CSA) if every maximal abelian subgroup of 𝐺 is malnormal. In this paper, as a natural generalization, we study groups with the property that all maximal nilpotent subgroups of class at most 𝑘 are malnormal, which we refer to as CSN𝑘 groups, and we show that they have many properties in common with the more widely studied CSA groups. In addition, we introduce the class of nilpotency transitive groups of class 𝑘, denoted NT𝑘, and in the presence of a special residuality condition, we prove that the CSN𝑘 and NT𝑘 properties are equivalent.