{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "031M8HX7DA"
    },
    {
      "field": "uses",
      "target": "04DC0SRN2K"
    },
    {
      "field": "uses",
      "target": "0736AXH2ME"
    },
    {
      "field": "uses",
      "target": "0BZVNWJ352"
    },
    {
      "field": "uses",
      "target": "0G0B3DRG65"
    },
    {
      "field": "uses",
      "target": "0JQ1KNQW2K"
    },
    {
      "field": "uses",
      "target": "0PCKBBMEXC"
    },
    {
      "field": "uses",
      "target": "0S7495H4HM"
    }
  ],
  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe order of $G$, written $|G|$, is the cardinality of the underlying set of $G$.\nWhen $G$ is finite, $|G|$ is the number of elements of $G$.\nWhen $G$ is infinite, $G$ has infinite order.\n\n## Remarks\n\nThe [[trivial group|0BNPC9850Y]] has order $1$, and the [[group of $n$th roots of unity|0WX11EA3F3]] has order $n$.\nThe group $(\\mathbb{Z}, +)$ has infinite order.\n[[Lagrange's theorem|3C53TD]] says that the order of a subgroup of a finite group divides the order of the group.\n\n## Explanations\n\n### L1 human\n\nThe order of a group tells you how big the group is.\nFor example, the rotations of a square that map the square to itself form a group of order $4$.\nThe word \"order\" has a second meaning in group theory: the order of an element.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0B85S2V2DR",
    "kind": "definition",
    "msc": [
      "20A05"
    ],
    "name": "Order of a group",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Order_(group_theory)"
      }
    ],
    "tag": "0SEX9F9J5Z",
    "uses": [
      "G5S1M9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0SEX9F9J5Z/",
  "links": [
    {
      "field": "defines",
      "target": "0B85S2V2DR"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0SEX9F9J5Z.md",
  "msc": [
    "20A05"
  ],
  "name": "Order of a group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0SE/0SEX9F9J5Z.md",
  "status": null,
  "tag": "0SEX9F9J5Z",
  "uri": "m2://0SEX9F9J5Z",
  "url": "https://math2.tamnd.com/wiki/order-of-a-group-definition/",
  "writing": {
    "tag": "0SEX9F9J5Z",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}