Solubility of groups can be characterized by configuration

TL;DR

This paper demonstrates that properties like solubility and FC of groups can be characterized by configuration sets, and introduces strong configuration equivalence, which coincides with isomorphism, providing new tools for group classification.

Contribution

It introduces the concept of configuration-based characterization for soluble and FC groups and defines strong configuration equivalence that aligns exactly with isomorphism.

Findings

Solubility and FC properties are characterized by configuration sets.

Strong configuration equivalence is shown to be equivalent to group isomorphism.

New criteria for group isomorphism based on configuration pairs.

Abstract

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. We show that group properties of being soluble or FC can be characterized by configuration sets. Then we investigate some condition on configuration pairs, which leads to isomorphism. We introduce a somewhat different notion of configuration equivalence, namely strong configuration equivalence, and prove that strong configuration equivalence coincides with isomorphism.