{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "WC16TBY3FN"
    }
  ],
  "body": "\n## Description\n\nThe Ree groups are two infinite families of finite [[groups of Lie type|NPEBABQ49M]], found by Rimhak Ree.\n\n1. For $q = 3^{2n+1}$ with $n \\ge 0$, the group ${}^2G_2(q)$ is the subgroup of $G_2(\\F_q)$ fixed by a twisted automorphism of characteristic $3$.\n2. For $q = 2^{2n+1}$ with $n \\ge 0$, the group ${}^2F_4(q)$ is the subgroup of $F_4(\\F_q)$ fixed by a twisted automorphism of characteristic $2$.\n\nEach twisted automorphism combines the exceptional graph automorphism of the Dynkin diagram with a field automorphism.\nTheir orders are\n$$\\abs{{}^2G_2(q)} = q^3 (q^3 + 1)(q - 1), \\qquad \\abs{{}^2F_4(q)} = q^{12} (q^6 + 1)(q^4 - 1)(q^3 + 1)(q - 1).$$\nFor $n \\ge 1$ these groups are non-abelian [[simple groups|0YMYMNQ0QZ]].\n\n## Remarks\n\nThe construction is possible because the algebraic group $G_2$ in characteristic $3$, and the algebraic group $F_4$ in characteristic $2$, has an exceptional endomorphism that exchanges long and short roots.\nThe smallest cases are not simple.\n${}^2G_2(3)$ has order $1512$ and its [[commutator subgroup|0ASGSTXSBW]] is $\\mathrm{SL}_2(\\F_8)$, of index $3$.\n${}^2F_4(2)$ has order $35942400$ and its commutator subgroup, of index $2$, is the [[Tits group|WC16TBY3FN]], which is simple.\nThe [[Suzuki groups|36X3GET13B]] arise from $B_2$ in characteristic $2$ in the same way.\n\n## Explanations\n\n### L1 human\n\nThe Ree groups are twisted versions of two exceptional matrix groups, $G_2$ and $F_4$, over special finite fields.\nThe twist is a symmetry that exists only in characteristic $3$ for $G_2$ and characteristic $2$ for $F_4$.\nThey give two more infinite families of finite simple groups.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "object",
    "msc": [
      "20D06",
      "20G40",
      "20G41"
    ],
    "name": "Ree groups",
    "same_as": {
      "wikidata": "Q7306554",
      "wikipedia": "Ree group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ree_group"
      }
    ],
    "tag": "YB62XEQJAC",
    "uses": [
      "NPEBABQ49M",
      "0YMYMNQ0QZ",
      "0B85S2V2DR"
    ]
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/YB62XEQJAC/",
  "links": [
    {
      "field": "uses",
      "target": "NPEBABQ49M"
    },
    {
      "field": "uses",
      "target": "0YMYMNQ0QZ"
    },
    {
      "field": "uses",
      "target": "0B85S2V2DR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YB62XEQJAC.md",
  "msc": [
    "20D06",
    "20G40",
    "20G41"
  ],
  "name": "Ree groups",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YB6/YB62XEQJAC.md",
  "status": null,
  "tag": "YB62XEQJAC",
  "uri": "m2://YB62XEQJAC",
  "url": "https://math2.tamnd.com/wiki/ree-groups/",
  "writing": {
    "tag": "YB62XEQJAC",
    "kind": "object",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}