Canonical and algebraically closed groups in universal classes of Abelian groups

Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes ${\mathcal{K}}$ within a quasivariety ${\mathfrak{A}}_p$, the class of groups whose periodic part is a $p$-group for a prime $p$. Also the concept of an algebraically closed group in ${\mathcal{K}}$ is introduced, and such groups are classified.