{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $G$ be a finite group and let $g \\in G$.\n2. The [[cyclic subgroup|0YT586TKXF]] $\\langle g \\rangle$ is a subgroup of $G$.\n3. [[The order of $\\langle g \\rangle$ is the order of $g$|031M8HX7DA]].\n4. By [[Lagrange's theorem|3C53TD]], $|\\langle g \\rangle|$ divides $|G|$.\n5. Thus the order of $g$ divides $|G|$.\n\n## Idea\n\nThe element generates a subgroup whose order is the order of the element. Lagrange's theorem then gives the result.\n",
  "commit": "f7cc1f60ee7d22cafedd3ef7b979a1abe7bcb828",
  "front": {
    "kind": "proof",
    "methods": [
      "0NTF6DMVAP"
    ],
    "msc": [
      "20D60"
    ],
    "name": "Proof that the order of an element divides the order of the group",
    "proves": "08V7JHS3QZ",
    "sources": [
      {
        "locator": "Applications",
        "url": "https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)"
      }
    ],
    "tag": "BMD35Z5SGC",
    "uses": [
      "0YT586TKXF",
      "031M8HX7DA",
      "3C53TD"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/BMD35Z5SGC/",
  "links": [
    {
      "field": "proves",
      "target": "08V7JHS3QZ"
    },
    {
      "field": "uses",
      "target": "0YT586TKXF"
    },
    {
      "field": "uses",
      "target": "031M8HX7DA"
    },
    {
      "field": "uses",
      "target": "3C53TD"
    },
    {
      "field": "methods",
      "target": "0NTF6DMVAP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/BMD35Z5SGC.md",
  "msc": [
    "20D60"
  ],
  "name": "Proof that the order of an element divides the order of the group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BMD/BMD35Z5SGC.md",
  "status": null,
  "tag": "BMD35Z5SGC",
  "uri": "m2://BMD35Z5SGC",
  "url": "https://math2.tamnd.com/wiki/proof-that-the-order-of-an-element-divides-the-order-of-the-group/",
  "writing": {
    "tag": "BMD35Z5SGC",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}