Elementary equivalence of the automorphism groups of reduced Abelian p-groups

TL;DR

This paper establishes a connection between the elementary equivalence of automorphism groups of unbounded reduced Abelian p-groups and the second-order logical equivalence of the groups themselves, based on their basic subgroups' ranks.

Contribution

It proves that elementary equivalence of automorphism groups implies second-order logical equivalence of the original groups, linking automorphism group properties to group structure.

Findings

Automorphism groups are elementarily equivalent if and only if the groups are second-order equivalent.

The equivalence depends on the final rank of the basic subgroups.

Results apply to unbounded reduced Abelian p-groups with p > 2.

Abstract

Consider unbounded reduced Abelian p-groups (p > 2) A and A'. In this paper, we prove that if the automorphism groups Aut A and Aut A' are elementary equivalent then the groups A and A' are equivalent in the second order logic bounded by the final rank of the basic subgroups of A and A'.