{
  "aliases": [
    "fundamental-theorem-of-finite-abelian-groups",
    "basis-theorem-for-finite-abelian-groups"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0MC0AQZSHE"
    },
    {
      "field": "uses",
      "target": "0RY0B72XAS"
    }
  ],
  "body": "\n## Statement\n\nLet $G$ be a finite [[abelian group|082GHNPJ53]].\nThen $G$ is [[isomorphic|08MG7H8JW5]] to a [[direct sum|00HF4RJ5QN]] $\\mathbb{Z}/q_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/q_t\\mathbb{Z}$, where each $q_i$ is a power of a prime number.\nThe numbers $q_1, \\ldots, q_t$ are unique up to their order.\nEquivalently, $G \\cong \\mathbb{Z}/k_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/k_u\\mathbb{Z}$ with $1 \u003c k_1 \\mid k_2 \\mid \\cdots \\mid k_u$, and these $k_i$ are unique.\n\n## Remarks\n\nThe two forms are equivalent because $\\mathbb{Z}/m n\\mathbb{Z} \\cong \\mathbb{Z}/m\\mathbb{Z} \\oplus \\mathbb{Z}/n\\mathbb{Z}$ if and only if $\\gcd(m, n) = 1$, as in [[this statement|08FDFEF8D1]].\nEach abelian group of order $8$ is isomorphic to exactly one of $\\mathbb{Z}/8\\mathbb{Z}$, $\\mathbb{Z}/4\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z}$ and $\\mathbb{Z}/2\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z}$.\nEach abelian group of order $15$ is cyclic, because $15 = 3 \\cdot 5$ with $\\gcd(3, 5) = 1$.\nThe theorem is the case of rank $0$ of the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]].\n\n## History\n\nLeopold Kronecker proved the theorem in 1870 with a group-theoretic proof, but he did not state it in the language of groups.\nHis work extended an earlier result of Carl Friedrich Gauss from 1801 on quadratic forms.\nFerdinand Georg Frobenius and Ludwig Stickelberger stated and proved the theorem in the language of groups in 1878.\n\n## Explanations\n\n### L1 human\n\nEach finite abelian group is built from cyclic groups.\nYou can choose the cyclic pieces to have prime power orders, or to have orders that divide each other.\nIn both forms, the pieces are unique, so this lists all finite abelian groups.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "fundamental-theorem-of-finite-abelian-groups",
      "basis-theorem-for-finite-abelian-groups"
    ],
    "kind": "statement",
    "msc": [
      "20K01"
    ],
    "name": "Classification of finite abelian groups",
    "role": "theorem",
    "sources": [
      {
        "locator": "Classification",
        "url": "https://en.wikipedia.org/wiki/Abelian_group"
      },
      {
        "locator": "Primary decomposition",
        "url": "https://en.wikipedia.org/wiki/Finitely_generated_abelian_group"
      },
      {
        "locator": "Invariant factor decomposition",
        "url": "https://en.wikipedia.org/wiki/Finitely_generated_abelian_group"
      },
      {
        "locator": "History",
        "url": "https://en.wikipedia.org/wiki/Finitely_generated_abelian_group"
      }
    ],
    "tag": "07CQR9BW3R",
    "uses": [
      "071DDFVG4J",
      "0B85Q41RP6",
      "085PND4KNT",
      "0YP7WYXF9V",
      "08FDFEF8D1"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/07CQR9BW3R/",
  "links": [
    {
      "field": "uses",
      "target": "071DDFVG4J"
    },
    {
      "field": "uses",
      "target": "0B85Q41RP6"
    },
    {
      "field": "uses",
      "target": "085PND4KNT"
    },
    {
      "field": "uses",
      "target": "0YP7WYXF9V"
    },
    {
      "field": "uses",
      "target": "08FDFEF8D1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/07CQR9BW3R.md",
  "msc": [
    "20K01"
  ],
  "name": "Classification of finite abelian groups",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07C/07CQR9BW3R.md",
  "status": null,
  "tag": "07CQR9BW3R",
  "uri": "m2://07CQR9BW3R",
  "url": "https://math2.tamnd.com/wiki/classification-of-finite-abelian-groups/",
  "writing": {
    "tag": "07CQR9BW3R",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}