Structure theorem for finitely generated abelian groups

Statement

Let $G$ be a finitely generated abelian group. Then $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$ for an integer $n \geq 0$ and prime powers $q_1, \ldots, q_t$. Also $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$. The rank $n$, the multiset of the $q_i$ and the sequence of the $k_i$ are unique.

Remarks

The finite part is the torsion subgroup of $G$, and $\mathbb{Z}^n$ is a free abelian group of rank $n$. So $G$ is finite if and only if $n = 0$, and then the theorem is the classification of finite abelian groups. Each finitely generated torsion-free abelian group is free abelian. The group $(\mathbb{Q}, +)$ shows that the condition "finitely generated" is necessary. The theorem is a special case of the structure theorem for finitely generated modules over a principal ideal domain.

History

Henry John Stephen Smith solved the case of finitely presented abelian groups in 1861, with the Smith normal form of integer matrices. Henri Poincaré proved the theorem for finitely generated abelian groups in 1900, with a matrix proof. Emmy Noether extended Kronecker's proof to finitely generated abelian groups in 1926.

Explanations

L1 human

A finitely generated abelian group is a free part $\mathbb{Z}^n$ plus a finite part. The finite part is a sum of cyclic groups. So a short list of integers describes the group completely.