A note on pseudofinite groups of finite centraliser dimension

TL;DR

This paper provides a structural characterization of pseudofinite groups with finite centraliser dimension and shows that no finitely generated pseudofinite group has this property.

Contribution

It introduces a structural theorem for pseudofinite groups of finite centraliser dimension and establishes the non-existence of finitely generated examples.

Findings

Structural theorem for pseudofinite groups of finite centraliser dimension

Finitely generated pseudofinite groups cannot have finite centraliser dimension

Clarifies the limitations of pseudofinite groups with respect to generation and centraliser properties

Abstract

We give a structural theorem for pseudofinite groups of finite centraliser dimension. As a corollary, we observe that there is no finitely generated pseudofinite group of finite centraliser dimension.