Countable-configurations and paradoxical decompositions

TL;DR

This paper introduces the concept of countable-configuration of groups, establishing that two Hopfian groups with identical configurations are isomorphic, and characterizes groups with countable paradoxical decompositions as precisely the infinite groups.

Contribution

It defines countable-configuration for groups and proves their isomorphism characterization among Hopfian groups, also linking paradoxical decompositions to infiniteness.

Findings

Two Hopfian groups with the same countable-configuration are isomorphic.

A group admits a countable paradoxical decomposition if and only if it is infinite.

The concept of countable-configuration distinguishes group structures.

Abstract

In this paper we define countable-configuration of groups and prove that two Hopfian groups with the same set of countable-configurations are isomorphic and vice versa. We also study the countable paradoxical decomposition of groups. It is proved that a group G admits a countable paradoxical decomposition if and only if it is infinite.