Pseudofinite abelian groups and asymptotic classes of finite abelian groups

This paper studies pseudofinite abelian groups and counting properties of definable sets in finite abelian groups and their ultraproducts. First, we characterize pseudofinite abelian groups in terms of Szmielew invariants and, building on results of Macintyre, extend this analysis to obtain a characterization of pseudofinite $\omega$-stable abelian groups. We then give an alternative proof that the class of finite abelian groups forms a multidimensional exact class, explicitly describing the possible cardinalities of definable sets in classes of finite abelian groups. We also investigate $N$-dimensional asymptotic classes of finite abelian groups and provide a series of examples showing that, despite the exactness result, several natural classes of finite abelian groups fail to be $N$-dimensional asymptotic classes for every $N\ge 1$. Finally, in an appendix, we characterize the SU-rank of pseudofinite abelian groups, building on work of Rogers. In particular, we show that for every pseudofinite abelian group $A$, either $\mathrm{Th}(A,+)$ is superstable of finite SU-rank or it is strictly stable.