First-order recognisability in finite and pseudofinite groups

TL;DR

This paper investigates the first-order properties of finite and pseudofinite groups, showing that certain characterizations like solubility are unique, while others like nilpotence and perfectness are not, and proves a weak Frattini-type result for pseudofinite groups.

Contribution

It demonstrates the uniqueness of first-order characterizations for solubility in finite groups and establishes a weak Frattini theorem for pseudofinite groups.

Findings

Solubility has a unique first-order characterization in finite groups.

Nilpotence and perfectness do not have such characterizations.

A weak Frattini's theorem holds for pseudofinite groups.

Abstract

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini's theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.