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  "body": "\n## Statement\n\nLet $G$ be a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].\nThen $G \\cong \\mathbb{Z}^n \\oplus \\mathbb{Z}/q_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/q_t\\mathbb{Z}$ for an integer $n \\geq 0$ and prime powers $q_1, \\ldots, q_t$.\nAlso $G \\cong \\mathbb{Z}^n \\oplus \\mathbb{Z}/k_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/k_u\\mathbb{Z}$ with $1 \u003c k_1 \\mid k_2 \\mid \\cdots \\mid k_u$.\nThe rank $n$, the multiset of the $q_i$ and the sequence of the $k_i$ are unique.\n\n## Remarks\n\nThe finite part is the [[torsion subgroup|0DBWXQRXQJ]] of $G$, and $\\mathbb{Z}^n$ is a [[free abelian group|0WJX8MGG37]] of rank $n$.\nSo $G$ is finite if and only if $n = 0$, and then the theorem is the [[classification of finite abelian groups|07CQR9BW3R]].\nEach finitely generated torsion-free abelian group is free abelian. The group $(\\mathbb{Q}, +)$ shows that the condition \"finitely generated\" is necessary.\nThe theorem is a special case of the structure theorem for finitely generated modules over a principal ideal domain.\n\n## History\n\nHenry John Stephen Smith solved the case of finitely presented abelian groups in 1861, with the Smith normal form of integer matrices.\nHenri Poincaré proved the theorem for finitely generated abelian groups in 1900, with a matrix proof.\nEmmy Noether extended Kronecker's proof to finitely generated abelian groups in 1926.\n\n## Explanations\n\n### L1 human\n\nA finitely generated abelian group is a free part $\\mathbb{Z}^n$ plus a finite part.\nThe finite part is a sum of cyclic groups.\nSo a short list of integers describes the group completely.\n",
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