Isotypical equivalence of periodic Abelian groups

TL;DR

This paper characterizes when periodic Abelian groups are isotypically equivalent using invariants, and explores the elementary equivalence of their divisible parts and basic subgroups, revealing structural homogeneity.

Contribution

It introduces invariants for isotypical classification of periodic Abelian groups and establishes criteria for their elementary equivalence and homogeneity.

Findings

Two Abelian p-groups with separable reduced parts are isotypically equivalent iff their divisible parts and basic subgroups are elementarily equivalent.

Provided invariants that distinguish isotypical classes of periodic Abelian groups.

Proved that such groups are ω-strongly homogeneous.

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.