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  "body": "\n## Summary\n\nLet $G$ be a finitely generated [[abelian group|082GHNPJ53]], and write $G \\cong \\mathbb{Z}^n \\oplus \\mathbb{Z}/k_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/k_u\\mathbb{Z}$ with $1 \u003c k_1 \\mid k_2 \\mid \\cdots \\mid k_u$.\nThe integers $k_1, \\ldots, k_u$ are the invariant factors of $G$.\nIn the other form $G \\cong \\mathbb{Z}^n \\oplus \\mathbb{Z}/q_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/q_t\\mathbb{Z}$, the prime powers $q_1, \\ldots, q_t$ are the elementary divisors of $G$.\n\n## Remarks\n\nThe [[structure theorem|0RY0B72XAS]] says that the rank $n$, the invariant factors and the elementary divisors are unique, and that they determine $G$ up to isomorphism.\nThe elementary divisors are the prime power factors of the invariant factors. For example, the invariant factors $2, 12$ give the elementary divisors $2, 4, 3$.\nFor a finite group $G$, the product of the invariant factors is $|G|$, and the largest invariant factor $k_u$ is the [[exponent|0ZDW0H0VX4]] of $G$.\nThe same terms are used for finitely generated modules over a principal ideal domain.\n\n## Explanations\n\n### L1 human\n\nThere are two standard ways to write a finitely generated abelian group as a sum of cyclic groups.\nThe invariant factors are orders that divide each other.\nThe elementary divisors are prime power orders.\n",
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