Isotypical equivalence of periodic Abelian groups (2402.11261v4)

Abstract: In this paper we give invariants that characterize isotypically equivalent Abelian periodic groups. Also, we describe types of standart tuples of elements in these groups. As the particular case we prove that two Abelian $p$-groups with separable reduced parts are isotypically equivalent if and only if their divisible parts and their basic subgroups are elementarily equivalent. Also as a corollary we prove that any Abelian $p$-group with a separable reduced part is $\omega$-strongly homogeneous.