Generic theories for series of finite Abelian groups

The notion of a generic theory $\mathsf{GTh}(\mathcal K,\mu)$ with respect to a measure $\mu$ was introduced in [Algebra i Logika, 53, No. 6, 779–789 (2014)]. Here, based on elementary invariants for Abelian groups and using a measure generated by a Frechet filter, we describe generic theories for two series of cyclic groups. Axioms of generic theories are given, complete theories are characterized in terms of elementary invariants, and canonical models of complete theories are constructed.