The dp-rank of abelian groups

TL;DR

This paper provides a formula for computing the dp-rank of abelian groups, relates it to VC density, characterizes strong abelian groups, and proves that infinite stable fields of finite dp-rank are algebraically closed.

Contribution

It introduces a method to compute the dp-rank of abelian groups and establishes new characterizations of strong abelian groups and properties of stable fields.

Findings

A formula for dp-rank of abelian groups is provided.

Dp-rank equals Vapnik-Chervonenkis density for one-based groups.

Infinite stable fields of finite dp-rank are algebraically closed.

Abstract

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik-Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups $A$ such that there is only finitely many primes $p$ such that the group $A/pA$ is infinite and for every prime $p$, there is only finitely many natural numbers $n$ such that $(p^nA)[p]/(p^{n+1}A)[p]$ is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.