The R_\infty property for abelian groups

Abstrakt

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups
do not have the $R_\infty$ property, but this does not hold for all of them! In fact we construct an uncountable number of infinite countable abelian groups
which do have the $R_{\infty}$ property.
We also construct an abelian group such that the cardinality of the Reidemeister classes is uncountable for any automorphism of that group.