Fundamental Theorem of Finitely Generated Abelian Groups

Fundamental Theorem of Finitely Generated Abelian Groups. Let $G$ be a finitely generated abelian group (i.e. $G$ has a finite generating set ). Then there exist integers $r\ge 0$ and $n_1,\dots,n_k \ge 2$ with $n_1 \mid n_2 \mid \cdots \mid n_k$ such that

where $\oplus$ denotes the direct sum of groups. The integer $r$ (the free rank) and the invariant factors $n_1,\dots,n_k$ are uniquely determined by $G$.

Equivalently, $G$ decomposes as a direct sum of cyclic groups of prime-power order (the “elementary divisor” form). This theorem is the group-theoretic specialization of the structure theorem for finitely generated modules over a PID with the PID $\mathbb{Z}$.

Proof sketch. One reduces to a presentation of $G$ as $\mathbb{Z}^n/R$ for a subgroup $R \le \mathbb{Z}^n$ and then diagonalizes the relations using integer row/column operations (Smith normal form). The diagonal entries give the invariant factors, and the number of zero diagonal entries gives the rank $r$.