{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n$ be a positive integer, and let $G$ be a [[primitive permutation group|Y10S0751KG]] on a set $\\Omega$ with $n$ elements.\nLet $p$ be a [[prime number|01826KMG2H]] with $p \\le n - 3$.\nSuppose that $G$ contains a [[cycle|07CT9RDY7Q]] of length $p$.\nThen $G$ is the [[alternating group|0Z5905Q73D]] $A_n$ or the [[symmetric group|0515A6D2Y1]] $S_n$ on $\\Omega$.\n\n## Remarks\n\nThe proof uses that the $p$-cycle fixes at least three points.\nCombining this with primitivity, one shows that $G$ is [[multiply transitive|BDJ1AX4XBB]], and then that $G$ contains $A_n$.\n\nThe bound $p \\le n - 3$ is necessary.\nThe cyclic group generated by $(1\\,2\\,3\\,4\\,5)$ is primitive on $5$ points and contains a $5$-cycle, so $p = n$, but it is neither $A_5$ nor $S_5$.\nThe group $\\mathrm{PGL}_2(\\F_5)$ of order $120$ acts primitively on the $6$ points of the projective line over $\\F_5$.\nThere the map $x \\mapsto x + 1$ is a $5$-cycle, so $p = n - 1$.\nBut $120$ is neither $360$ nor $720$, so the group is neither $A_6$ nor $S_6$.\nThe group $\\mathrm{PSL}_2(\\F_8)$ of order $504$ acts $2$-transitively, hence primitively, on the $9$ points of the projective line over $\\F_8$.\nFor a generator $a$ of $\\F_8^\\times$, the map $x \\mapsto ax$ fixes $0$ and $\\infty$ and is a $7$-cycle, so $p = n - 2$.\nBut $504$ is smaller than $\\abs{A_9}$, so the group is neither $A_9$ nor $S_9$.\n\nThe theorem gives a practical test for a permutation group of degree $n$ to contain $A_n$.\nIt is enough that the group is primitive and contains one cycle of prime length at most $n - 3$.\n\n## Explanations\n\n### L1 human\n\nPrimitive groups shuffle a set without respecting any grouping of its points.\nJordan's theorem says that such a group is very large as soon as it contains one special element.\nThis element is a cycle of prime length that leaves at least three points in place.\nThen the group contains all even permutations.\nSo a single element, together with primitivity, can force a group to be huge.\n\n### L2 human\n\nThe theorem is a classical tool to prove that a given permutation group is the full symmetric or alternating group.\nIt is used, for example, to show that the Galois group of a polynomial is $S_n$.\nAn irreducible polynomial gives a transitive Galois group, and one then looks for suitable cycles.\nSimilar results say that a primitive group containing a transposition is $S_n$, and a primitive group containing a $3$-cycle contains $A_n$.\nThe proof works with [[multiply transitive group actions|BDJ1AX4XBB]] and the sets of fixed points of elements.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "Y10S0751KG",
      "0515A6D2Y1",
      "0Z5905Q73D"
    ],
    "kind": "statement",
    "msc": [
      "20B15",
      "20B20",
      "20B30"
    ],
    "name": "Jordan's theorem on primitive groups containing a prime cycle",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6276298",
      "wikipedia": "Jordan's theorem (symmetric group)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Jordan's_theorem_(symmetric_group)"
      }
    ],
    "tag": "78R2TV41D4",
    "uses": [
      "Y10S0751KG",
      "07CT9RDY7Q",
      "01826KMG2H",
      "0515A6D2Y1",
      "0Z5905Q73D",
      "BDJ1AX4XBB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/78R2TV41D4/",
  "links": [
    {
      "field": "about",
      "target": "Y10S0751KG"
    },
    {
      "field": "about",
      "target": "0515A6D2Y1"
    },
    {
      "field": "about",
      "target": "0Z5905Q73D"
    },
    {
      "field": "uses",
      "target": "Y10S0751KG"
    },
    {
      "field": "uses",
      "target": "07CT9RDY7Q"
    },
    {
      "field": "uses",
      "target": "01826KMG2H"
    },
    {
      "field": "uses",
      "target": "0515A6D2Y1"
    },
    {
      "field": "uses",
      "target": "0Z5905Q73D"
    },
    {
      "field": "uses",
      "target": "BDJ1AX4XBB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/78R2TV41D4.md",
  "msc": [
    "20B15",
    "20B20",
    "20B30"
  ],
  "name": "Jordan's theorem on primitive groups containing a prime cycle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/78R/78R2TV41D4.md",
  "status": null,
  "tag": "78R2TV41D4",
  "uri": "m2://78R2TV41D4",
  "url": "https://math2.tamnd.com/wiki/jordans-theorem-on-primitive-groups-containing-a-prime-cycle/",
  "writing": {
    "tag": "78R2TV41D4",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}