{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a finite [[simple group|0YMYMNQ0QZ]].\nThe group $G$ is *sporadic* when it is not isomorphic to any of the following:\n\n1. a [[cyclic group|0KVGH0J5B3]] of prime order;\n2. an [[alternating group|0Z5905Q73D]] $A_n$ with $n \\ge 5$;\n3. a simple [[group of Lie type|NPEBABQ49M]].\n\n## Remarks\n\nBy the [[classification of finite simple groups|06J6P03MG7]], there are exactly $26$ sporadic groups up to isomorphism.\nThe [[Tits group|WC16TBY3FN]] ${}^2F_4(2)'$ is usually counted with the groups of Lie type in clause 3; some authors count it as a $27$th sporadic group.\nExample: the [[Mathieu group|T1KR4KPX8P]] $M_{11}$ is sporadic.\nIt is simple and not cyclic, its order $7920$ is not of the form $n!/2$, and no simple group of Lie type has order $7920$.\nNon-example: $A_5$ is simple but not sporadic, since it is an alternating group, so clause 2 fails.\nThe sporadic groups include the five Mathieu groups, the four [[Janko groups|PSEC8D7CX0]], the three [[Conway groups|366D4ETGQH]] and the three [[Fischer groups|AYJ1CM1XT8]].\nThe others are the Higman–Sims, McLaughlin, Held, Rudvalis, Suzuki, O'Nan, Harada–Norton, Lyons and Thompson groups, the [[Baby Monster|TNPQ9AWVBC]] and the [[Monster|C9VY20TAPT]].\n\n## Explanations\n\n### L1 human\n\nA sporadic group is a finite simple group that does not belong to the regular infinite families.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "PZ49Z4CBFZ",
    "kind": "definition",
    "msc": [
      "20D08",
      "20D05"
    ],
    "name": "Sporadic group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sporadic_group"
      }
    ],
    "tag": "390Z8QV5ZG",
    "uses": [
      "0YMYMNQ0QZ",
      "0KVGH0J5B3",
      "0Z5905Q73D",
      "NPEBABQ49M"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/390Z8QV5ZG/",
  "links": [
    {
      "field": "defines",
      "target": "PZ49Z4CBFZ"
    },
    {
      "field": "uses",
      "target": "0YMYMNQ0QZ"
    },
    {
      "field": "uses",
      "target": "0KVGH0J5B3"
    },
    {
      "field": "uses",
      "target": "0Z5905Q73D"
    },
    {
      "field": "uses",
      "target": "NPEBABQ49M"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/390Z8QV5ZG.md",
  "msc": [
    "20D08",
    "20D05"
  ],
  "name": "Sporadic group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/390/390Z8QV5ZG.md",
  "status": null,
  "tag": "390Z8QV5ZG",
  "uri": "m2://390Z8QV5ZG",
  "url": "https://math2.tamnd.com/wiki/sporadic-group-definition/",
  "writing": {
    "tag": "390Z8QV5ZG",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}