Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups

TL;DR

This paper characterizes when the endomorphism rings and automorphism groups of periodic Abelian groups are elementarily equivalent, based on properties of their p-components and second-order logic equivalences.

Contribution

It establishes precise criteria linking elementary equivalence of endomorphism rings and automorphism groups to properties of p-components and second-order logic.

Findings

Endomorphism rings are elementarily equivalent iff p-components are.

Automorphism groups are elementarily equivalent under specific conditions.

Automorphism groups and endomorphism rings are equivalent when p-components are.

Abstract

In this paper, we prove that the endomorphism rings End A and End A' of periodic infinite Abelian groups A and A' are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all primes p. Additionally, we show that the automorphism groups Aut A and Aut A' of periodic Abelian groups A and A' that do not have 2-components and do not contain cocyclic p-components are elementarily equivalent if and only if, for any prime p, the corresponding p-components A_p and A_p' of A and A' are equivalent in second-order logic if they are not reduced, and are equivalent in second-order logic bounded by the cardinalities of their basic subgroups if they are reduced. For such groups A and A', their automorphism groups are elementarily equivalent if and only if their endomorphism rings are elementarily equivalent, and the automorphism groups of the corresponding p-components for all primes p are elementarily equivalent.