{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be an [[abelian|082GHNPJ53]] [[group|G5S1M9]].\nThen $G$ is [[simple|0YMYMNQ0QZ]] if and only if $G$ is finite of prime order.\nIn that case, $G$ is [[cyclic|0KVGH0J5B3]], and $G \\cong \\mathbb{Z}/p\\mathbb{Z}$ for a prime number $p$.\n\n## Remarks\n\nIn an abelian group, each subgroup is normal, so a simple abelian group has no subgroup except $\\{e\\}$ and itself.\nThen $G = \\langle g \\rangle$ for each $g \\neq e$, so $G$ is cyclic; $\\mathbb{Z}$ is not simple, because $2\\mathbb{Z}$ is a proper nontrivial subgroup.\nA finite cyclic group of order $n$ has a subgroup of order $d$ for each divisor $d$ of $n$, so $n$ must be prime.\n\n## Explanations\n\n### L1 human\n\nIn an abelian group, all subgroups are normal.\nSo a simple abelian group has no subgroups at all except the trivial ones.\nOnly the groups of prime order have this property.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "20E32",
      "20K01"
    ],
    "name": "Abelian simple groups have prime order",
    "role": "proposition",
    "sources": [
      {
        "locator": "Examples",
        "url": "https://en.wikipedia.org/wiki/Simple_group"
      },
      {
        "locator": "Finite simple groups",
        "url": "https://en.wikipedia.org/wiki/Simple_group"
      }
    ],
    "tag": "0JJV80RXHD",
    "uses": [
      "0C093D1T4M",
      "0B85Q41RP6",
      "0JM7PA288B",
      "0KB1ND7XDS"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0JJV80RXHD/",
  "links": [
    {
      "field": "uses",
      "target": "0C093D1T4M"
    },
    {
      "field": "uses",
      "target": "0B85Q41RP6"
    },
    {
      "field": "uses",
      "target": "0JM7PA288B"
    },
    {
      "field": "uses",
      "target": "0KB1ND7XDS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JJV80RXHD.md",
  "msc": [
    "20E32",
    "20K01"
  ],
  "name": "Abelian simple groups have prime order",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JJ/0JJV80RXHD.md",
  "status": null,
  "tag": "0JJV80RXHD",
  "uri": "m2://0JJV80RXHD",
  "url": "https://math2.tamnd.com/wiki/abelian-simple-groups-have-prime-order/",
  "writing": {
    "tag": "0JJV80RXHD",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}