Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 11G / b / Solution

Present a finitely generated abelian group $G$ as the cokernel of an integer matrix $A:{\mathbb{Z}}^m\to {\mathbb{Z}}^n$. Changing $A$ by the invertible row and column operations used in its Smith normal form does not change the isomorphism type of the cokernel. Ifthenwhere factors with $d_i=1$ may be omitted. This is the Fundamental theorem of finitely generated abelian groups; the rank of an abelian group $r$ and invariant factors $d_i>1$ are unique.