Configuration of nilpotent groups and isomorphism

TL;DR

This paper investigates how the concept of configuration in groups relates to their structure, particularly focusing on nilpotent groups, normal subgroups, and isomorphism conditions.

Contribution

It establishes conditions under which groups with the same configuration sets have related quotient structures, especially for nilpotent and FC-groups.

Findings

Groups with identical configurations have isomorphic quotient structures under certain conditions.

Configuration sets can determine structural properties of nilpotent and FC-groups.

The paper extends the understanding of how configurations relate to group isomorphism.

Abstract

The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied.