{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "05MM4EK4SA"
    },
    {
      "field": "uses",
      "target": "08V7JHS3QZ"
    },
    {
      "field": "uses",
      "target": "09JZ9P4899"
    },
    {
      "field": "uses",
      "target": "0GKQKF0FXJ"
    },
    {
      "field": "uses",
      "target": "0JM7PA288B"
    },
    {
      "field": "uses",
      "target": "0PDBVPP233"
    },
    {
      "field": "uses",
      "target": "0S19CWFZM2"
    },
    {
      "field": "uses",
      "target": "856B6C7EE4"
    },
    {
      "field": "uses",
      "target": "BMD35Z5SGC"
    },
    {
      "field": "uses",
      "target": "FESTJ0K9TX"
    },
    {
      "field": "illustrates",
      "target": "4DWC4XFS34"
    },
    {
      "field": "illustrates",
      "target": "6VFF2TTXYZ"
    },
    {
      "field": "proves",
      "target": "W44XETYC4Q"
    }
  ],
  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] and let $H$ be a subgroup of $G$.\nThen the order of $H$ divides the order of $G$.\n\n## Remarks\n\nThe quotient $|G| / |H|$ is the number of left cosets of $H$ in $G$.\n\n## Explanations\n\n### L1 agent\n\nThe left cosets of a subgroup cut the group into pieces of the same size.\nEach piece has as many elements as the subgroup.\nThus the size of the group is a whole number of copies of the size of the subgroup.\nFor example, a group with $12$ elements can have subgroups of order $1, 2, 3, 4, 6$ or $12$, and of no other order.\n\n### L2 agent\n\nLagrange's theorem gives a strong limit on the subgroups of a finite group. The order of each subgroup is a divisor of the order of the group.\n\nThe idea of the proof is to count cosets:\n\n1. The left cosets $aH$ of $H$ partition $G$.\n2. Each left coset has exactly $|H|$ elements, because $h \\mapsto ah$ is a bijection from $H$ to $aH$.\n3. If there are $k$ cosets, then $|G| = k\\,|H|$.\n\nThe number $k$ is the index $[G : H]$. Thus $|G| = [G : H] \\cdot |H|$.\n\nA worked example with $G = \\mathbb{Z}/12\\mathbb{Z}$ and $H = \\{0, 4, 8\\}$:\n\n1. $H$ is a subgroup of order $3$.\n2. The cosets are $H$, $1 + H = \\{1, 5, 9\\}$, $2 + H = \\{2, 6, 10\\}$ and $3 + H = \\{3, 7, 11\\}$.\n3. There are $4$ cosets of $3$ elements each, and $4 \\cdot 3 = 12$.\n\nTwo standard consequences are that the order of each element divides $|G|$, and that a group of prime order is cyclic.\n\nA common error is to believe the converse. A divisor $d$ of $|G|$ need not be the order of a subgroup. The alternating group $A_4$ has order $12$, but it has no subgroup of order $6$. The converse is true for some classes of groups, for example for cyclic groups and for groups of prime power order.\n\nThe theorem needs $G$ to be finite. For an infinite group the index can still be finite, as for $2\\mathbb{Z}$ in $\\mathbb{Z}$, but the orders are not numbers.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "G5S1M9",
      "0X1JXT47GA",
      "03DEH43F4H",
      "0B85S2V2DR"
    ],
    "formal": [
      {
        "adapter": "Math2.Match.lagrange",
        "decl": "Subgroup.card_subgroup_dvd_card",
        "library": "mathlib",
        "match": "stronger",
        "system": "lean4"
      }
    ],
    "inventory": [
      "ADBP6KTNFC"
    ],
    "kind": "statement",
    "msc": [
      "20D60"
    ],
    "name": "Lagrange's theorem",
    "role": "theorem",
    "sources": [
      {
        "book": "isbn:9780471433347",
        "locator": "Section 3.2, Theorem 8"
      }
    ],
    "tag": "3C53TD",
    "uses": [
      "G5S1M9"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/3C53TD/",
  "links": [
    {
      "field": "about",
      "target": "G5S1M9"
    },
    {
      "field": "about",
      "target": "0X1JXT47GA"
    },
    {
      "field": "about",
      "target": "03DEH43F4H"
    },
    {
      "field": "about",
      "target": "0B85S2V2DR"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/3C53TD.md",
  "msc": [
    "20D60"
  ],
  "name": "Lagrange's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3C5/3C53TD.md",
  "status": null,
  "tag": "3C53TD",
  "uri": "m2://3C53TD",
  "url": "https://tamnd.github.io/math2/wiki/lagranges-theorem/",
  "writing": {
    "tag": "3C53TD",
    "kind": "statement",
    "level": "W2",
    "next": [
      {
        "what": "section History",
        "note": "who stated it first and who proved it first",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanations for the audience",
        "have": 2,
        "want": 3
      },
      {
        "what": "a person confirmed each agent-written section",
        "note": "rule P04",
        "have": 0,
        "want": 1
      }
    ]
  }
}