On Stabilization of $E_k$ Chains

TL;DR

This paper investigates conditions under which certain subgroup chains in infinite groups stabilize, focusing on generalizations of double centralizers and their behavior in groups with chain conditions on centralizers.

Contribution

It introduces criteria for the finite stabilization of $E_k$ subgroup chains and explores their occurrence in groups satisfying chain conditions on centralizers.

Findings

Criteria for stabilization of $E_k$ chains established

Conditions under which chains terminate finitely identified

Analysis of chains in groups with chain condition on centralizers

Abstract

We study special subgroups of infinite groups that generalize double centralizers. We analyze sufficient conditions for descending chains of such subgroups to stop after finitely many steps. We discuss whether this phenomenon can happen in the class of groups satisfying chain condition on centralizers.