What is this method called - Abelization of a Group.
This is called computing the Smith normal form, and the application you describe features in this question: Determining the Smith Normal Form
I think this is an Application of the main structure theorem of finitely generated modules over a P.I.D applied to the case of abelian groups. If you have Jacobson algebra book go to page 188 (Ch. 3.10 "application to abelian groups and to linear transformations") I don't see a specific name for this method here, but it is well explained.