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A finitely generated abelian group has a finite generating set. Equivalently, it is the cokernel of a homomorphism between finitely generated free abelian groups.

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.

Let $T$ be the surface with boundary obtained by deleting the interiors of the $n$ disks. By excision,so $H_2(S,T)\cong {\mathbb{Z}}^n$ and the other relative groups vanish. In the long exact sequence in relative homology, the mapsends an orientation class to $(1,\dots ,1)$, up to an overall choice of sign. It is therefore injective and has cokernel ${\mathbb{Z}}^{n-1}$. Exactness gives $H_2(T)=0$ and a short exact sequenceThis sequence splits because its right-hand group is a free abelian group. Since $T$ is connected, we obtain