On the norm of the centralizers of a group

TL;DR

This paper characterizes finitely generated nilpotent groups using a series of centralizer-based subgroups, providing a new criterion for nilpotency in group theory.

Contribution

It introduces a novel characterization of finitely generated nilpotent groups through the properties of the centralizer-based series Cn(G).

Findings

Finitely generated groups are nilpotent iff G = Cn(G) for some n.

Provides a new criterion for identifying nilpotent groups.

Establishes a link between centralizer series and group nilpotency.

Abstract

For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper, we show that a finitely generated group G is nilpotent if and only if G = Cn(G) for some positive integer n.