Fundamental Theorem of Finitely Generated Abelian Groups

Fundamental Theorem of Finitely Generated Abelian Groups. Let $G$ be a finitely generated abelian group. Then there exist integers $r\ge 0$ and $n_1,\dots,n_k\ge2$ , with $n_1\mid n_2\mid\cdots\mid n_k$ , such that

G \cong \mathbb{Z}^r \oplus \mathbb{Z}/n_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/n_k\mathbb{Z},

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

Equivalent characterizations

Equivalently, the finite torsion subgroup of $G$ decomposes as a direct sum of cyclic groups of prime-power order. In this elementary-divisor form, the rank $r$ and the multiset of prime powers are uniquely determined by $G$ .

Remarks

This theorem is the group-theoretic specialization of the structure theorem for finitely generated modules over a PID with the PID $\mathbb{Z}$ .