On the power-commutative kernel of locally nilpotent groups

A group G is called power commutative, or a PC-group, if [xm, yn] = 1 implies [x, y] = 1 for all x, y ∈G such that xm = 1, yn = 1. So power-commutative groups are those groups in which commutativity of nontrivial powers of two elements implies commutativity of the two elements. Clearly, G is a PC-group if and only if CG(x) = CG(x) for all x ∈ G and all integers n such that xn = 1. Obvious examples of PC-groups are groups in which commutativity is a transitive relation on the set of nontrivial elements (CT-groups) and groups of prime exponent. Recall that a group G is called an R-group if xn = yn implies x = y for all x, y ∈ G and for all positive integers n. In other words, R-groups are groups in which the extraction of roots is unique. A result due to Mal’cev and Cernikov (see, e.g., [3]) states that every nilpotent torsion-free group is an R-group. There is a natural connection between PCgroups and R-groups. For, as pointed out in [3], a torsion-free group is a PC-group if and only if it is an R-group. In [5], Wu gave the classification of locally finite PC-groups. In particular, she proved that a finite group is a PC-group if and only if the centralizer of each nontrivial element is abelian or of prime exponent. This result implies that a finite group having a nontrivial center is a PC-group if and only if it is abelian or it has prime exponent. Moreover, the class of PC-groups is contained in the class of groups in which the centralizer of each nontrivial element is nilpotent. This class of groups was investigated by many authors (see, e.g., [1, 4]). In analogy to what is done in [2] to define the commutative-transitive kernel of a group, we introduce an ascending series