The DP-Rank of Abelian Groups

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 are only finitely many primes p such that the group A / pA is infinite and for every prime p , there are only finitely many natural numbers n such that $\\left( {p^n A} \\right)[p]/\\left( {p^{n + 1} A} \\right)[p]$\n is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.