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THE DP-RANK OF ABELIAN GROUPS
Published online by Cambridge University Press: 01 February 2019
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 $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.
Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.
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- Copyright © The Association for Symbolic Logic 2019
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