Definable envelopes of nilpotent subgroups of groups with chain conditions on centralizers

Abstract

An $\mathfrak{M}_C$ group is a group in which all chains of centralizers have finite length. In this article, we show that every nilpotent subgroup of an $\mathfrak{M}_C$ group is contained in a definable subgroup which is nilpotent of the same nilpotence class. Definitions are uniform when the lengths of chains are bounded.