Torsion subgroups of elementary equivalent abelian groups are elementarily equivalent

Statement
Suppose $$G$$ and $$H$$ are fact about::abelian groups that are elementarily equivalent. Then, their fact about::torsion subgroups, i.e., the subgroups comprising the elements that have finite order (also called periodic elements or torsion elements) are also elementarily equivalent groups.

Related facts

 * Quotients of elementarily equivalent abelian groups by multiples of n are elementarily equivalent
 * Finite groups are elementarily equivalent iff they are isomorphic
 * Finitely generated abelian groups are elementarily equivalent iff they are isomorphic