Abstract: 

The main results of the article concern the model theory of $T$-pseudofinite acts over abelian groups, where $T$ is the theory of all acts over the  group. A left act over a group  $G$ is a set  on which $G$ acts unitarily from the left. We give characteristic properties of an  abelian group $G$ that are necessary for the existence of a $T$-pseudofinite $G$-act in the class $K_{G,\bar S}$ and sufficient for the class $K_{G,\bar S}$ to be $T$-pseudofinite, where $T$ is the theory of all $G$-acts, $\bar S$ is a finite set  of subgroups of $G$, and  $K_{G,\bar S}$ is the class of all coproducts of $G$-acts of the form $_G(G/G_1)$, $G_1\in\bar S$. It follows from this result that for a finitely generated abelian group $G$ the class $K_{G,\bar S}$ is $T$-pseudofinite; for the multiplicative group of rational numbers $G$ the class $K_{G,\bar S}$ is also $T$-pseudofinite. It is noted that there exists an abelian group $G$ and a $G$-act $_GA$ such that $_GA$ is  not $T$-pseudofinite but is pseudofinite, where $T$ is the theory of all $G$-acts, namely, for a quasicyclic group $G$ as a divisible group, the $G$-act $_GG$ is not $T$-pseudofinite, but is pseudofinite.