# Finitely generated abelian groups are elementarily equivalent iff they are isomorphic

From Groupprops

## Statement

Suppose and are Finitely generated abelian group (?)s. Then, and are Elementarily equivalent groups (?) if and only if they are Isomorphic groups (?).

## Facts used

- Quotients of elementarily equivalent abelian groups by multiples of n are elementarily equivalent
- Structure theorem for finitely generated abelian groups
- Finite groups are elementarily equivalent iff they are isomorphic
- Torsion subgroups of elementary equivalent abelian groups are elementarily equivalent

## Proof

By fact (1), we have that, for every , and are elementarily equivalent. By fact (2), we have that:

where is a finite abelian group.

where is a finite abelian group.

First, choose as an integer greater than that is a multiple of the exponents of both and . Then, and are elementarily equivalent, and these are:

.

These are both finite groups, so by fact (3), they are isomorphic, so .

Further, by fact (4), the torsion subgroups of and are elementarily equivalent, so, since they are finite, fact (3) yields that .

Thus, .