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, .