Finite rank and pseudofinite groups

TL;DR

This paper proves that infinite finitely generated groups with finite Pr"ufer rank cannot be elementarily equivalent to ultraproducts of finite groups and are not pseudofinite, highlighting limitations in their model-theoretic properties.

Contribution

It establishes new results connecting finite Pr"ufer rank, pseudofiniteness, and elementary equivalence in finitely generated groups.

Findings

Infinite finitely generated groups are not elementarily equivalent to ultraproducts of finite groups of fixed Pr"ufer rank.

Finitely generated groups of finite Pr"ufer rank are not pseudofinite.

The results clarify the model-theoretic limitations of such groups.

Abstract

It is proven that an infinite finitely generated group cannot be elementarily equivalent to an ultraproduct of finite groups of a given Pr\"ufer rank. Furthermore, it is shown that an infinite finitely generated group of finite Pr\"ufer rank is not pseudofinite.