{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nBN-pairs can be used to prove that many groups of Lie type are [[simple|0C093D1T4M]] modulo their [[centers|0DZ93493VN]]. More precisely, if *G* has a *BN*-pair such that *B* is a [[solvable group|0FXJ7YFMMX]], the intersection of all conjugates of *B* is trivial, and the set of generators of *W* cannot be decomposed into two [[non-empty|0ECVK59Y03]] commuting sets, then *G* is simple whenever it is a perfect group. In practice all of these conditions except for *G* being perfect are easy to check. Checking that *G* is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showing that a group is perfect is usually far easier than showing it is simple.\n",
  "commit": "a799a9852bcfeae9dc83faa3bef60208bb53686b",
  "front": {
    "inventory": [
      "1YXNN0CNS5"
    ],
    "kind": "statement",
    "name": "Tits's simplicity theorem",
    "role": "theorem",
    "sources": [
      {
        "locator": "(B, N) pair, section Properties \u003e Tits's simplicity theorem",
        "mention": "wikipedia://en/1772133#8BPD",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:1772133@r1346225273"
      }
    ],
    "tag": "DNE3N4DCC5"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/DNE3N4DCC5/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/DNE3N4DCC5.md",
  "msc": null,
  "name": "Tits's simplicity theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DNE/DNE3N4DCC5.md",
  "status": null,
  "tag": "DNE3N4DCC5",
  "uri": "m2://DNE3N4DCC5",
  "url": "https://math2.tamnd.com/wiki/titss-simplicity-theorem/",
  "writing": {
    "tag": "DNE3N4DCC5",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}