Locally nilpotent groups containing an element that commutes with only finitely many of its conjugates

Abstract

We prove that a locally nilpotent group containing an element that commutes with only finitely many of its conjugates includes an abelian normal subgroup. We find some necessary conditions for the normal closure of such an element to be nilpotent.Keywords: locally nilpotent group, nilpotent group, abelian normal subgroup, normal closure of an element in a group, element commuting with only finitely many of its conjugates In this paper we continue the study that was initiated in [1, 2] of locally nilpotent groups containing an element that commutes with only finitely many of its conjugates.Theorem 1. If a nonidentity element a of a locally nilpotent group G commutes with only finitely many of its conjugates then the center of the normal closure a G is nontrivial. Corollary 1. A locally nilpotent group G containing a nonidentity element a that commutes with only finitely many of its conjugates has a nonidentity abelian normal subgroup. Theorem 2. An element of a torsion-free locally nilpotent group commutes with only finitely many of its conjugates if and only if it lies in the center of the group.Theorem 3. Let an element a of a locally nilpotent group G be of finite order and commute with only finitely many of its conjugates, and let the orders of all commutators of the form a −1 g −1 ag be bounded by the same constant. Then the normal closure a G is a nilpotent group.