Model theory of finite and pseudofinite groups

TL;DR

This survey explores the structure of pseudofinite groups using stability theory, aiming to connect model-theoretic insights with finite group theory for both group theorists and model theorists.

Contribution

It provides a comprehensive overview of how stability theory applies to pseudofinite groups and discusses potential implications for finite group theory.

Findings

Pseudofinite groups exhibit stability-theoretic properties.

Model theory offers new perspectives on the structure of finite groups.

The survey highlights open problems and future research directions.

Abstract

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.