Definable Envelopes of Nilpotent Subgroups of Groups with Chain Conditions on Centralizers

TL;DR

This paper proves that in groups with finite-length chains of centralizers, every nilpotent subgroup is contained in a definable nilpotent subgroup of the same class, with uniform definitions when chain lengths are bounded.

Contribution

It establishes the existence of definable envelopes for nilpotent subgroups in $ ext{M}_C$ groups, extending structural understanding of these groups.

Findings

Every nilpotent subgroup is contained in a definable nilpotent subgroup.

Definability is uniform when chain lengths are bounded.

Results apply to groups with finite chains of 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.