The DP-Rank of Abelian Groups

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 (pⁿA)[p]/(pⁿ⁺¹A)[p] is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.