{
  "aliases": [
    "power-of-the-group-order-is-identity"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] with identity element $e$, and let $g \\in G$.\nThen $g^{|G|} = e$.\n\n## Remarks\n\nThe [[order|0WE2NHZJ7B]] of $g$ divides $|G|$ by [[this corollary|08V7JHS3QZ]], and [[a power is $e$ when the order divides the exponent|0F5QC101Q7]].\nFor the group $(\\mathbb{Z}/p\\mathbb{Z})^\\times$ of order $p - 1$, with $p$ prime, this gives Fermat's little theorem $a^{p-1} \\equiv 1 \\pmod p$ for $p \\nmid a$.\nFor the group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ of order $\\varphi(n)$, it gives Euler's theorem.\n\n## Explanations\n\n### L1 human\n\nIf a group has $n$ elements, then the $n$th power of each element is the identity.\nFor example, in a group of order $6$, each element $g$ satisfies $g^6 = e$.\nThis simple fact proves Fermat's little theorem in number theory.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "power-of-the-group-order-is-identity"
    ],
    "kind": "statement",
    "msc": [
      "20D60"
    ],
    "name": "g to the order of the group is the identity",
    "role": "corollary",
    "sources": [
      {
        "locator": "Applications",
        "url": "https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory)"
      }
    ],
    "tag": "0ZEPNS9BXK",
    "uses": [
      "08V7JHS3QZ",
      "0F5QC101Q7"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0ZEPNS9BXK/",
  "links": [
    {
      "field": "uses",
      "target": "08V7JHS3QZ"
    },
    {
      "field": "uses",
      "target": "0F5QC101Q7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0ZEPNS9BXK.md",
  "msc": [
    "20D60"
  ],
  "name": "g to the order of the group is the identity",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0ZE/0ZEPNS9BXK.md",
  "status": null,
  "tag": "0ZEPNS9BXK",
  "uri": "m2://0ZEPNS9BXK",
  "url": "https://math2.tamnd.com/wiki/g-to-the-order-of-the-group-is-the-identity/",
  "writing": {
    "tag": "0ZEPNS9BXK",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}