{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $N = N_G(P)$ be the [[normaliser|0AT1ZSNM25]] of $P$ in $G$. It is a subgroup of $G$: if $gPg^{-1} = P$ and $hPh^{-1} = P$, then $(gh)P(gh)^{-1} = P$ and $g^{-1}Pg = P$.\n2. For $x \\in P$ we have $xPx^{-1} = P$, because $P$ is a subgroup. So $P \\subseteq N \\subseteq G$.\n3. By the [[index formula|04DC0SRN2K]] for $P$ in $G$, for $N$ in $G$, and for $P$ in $N$, we get $|G| = [G : P]\\,|P|$, $|G| = [G : N]\\,|N|$ and $|N| = [N : P]\\,|P|$.\n4. Combining these, $[G : P]\\,|P| = [G : N]\\,[N : P]\\,|P|$. Dividing by $|P| \\geq 1$ gives $[G : P] = [G : N] \\cdot [N : P]$.\n5. By the [[formula for the number of Sylow subgroups|09ZJ6CAK7S]], $n_p = [G : N]$.\n6. So $[G : P] = n_p \\cdot [N : P]$, where $[N : P]$ is a positive integer. Hence $n_p$ divides the [[index|04QEBF8S49]] $[G : P]$.\n\n## Idea\n\nThe number $n_p$ is the index of the normaliser of $P$. The normaliser sits between $P$ and $G$, and indices multiply along a chain of subgroups. So $n_p$ is a factor of $[G : P]$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "20D20"
    ],
    "name": "Proof that the number of Sylow subgroups divides the index",
    "proves": "0M3PYJ3TSE",
    "sources": [
      {
        "locator": "Proof of the Sylow theorems",
        "url": "https://en.wikipedia.org/wiki/Sylow_theorems"
      }
    ],
    "tag": "MRH07WB74D",
    "uses": [
      "0JWZX6FKSH",
      "0AT1ZSNM25",
      "09ZJ6CAK7S",
      "04DC0SRN2K",
      "04QEBF8S49"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/MRH07WB74D/",
  "links": [
    {
      "field": "proves",
      "target": "0M3PYJ3TSE"
    },
    {
      "field": "uses",
      "target": "0JWZX6FKSH"
    },
    {
      "field": "uses",
      "target": "0AT1ZSNM25"
    },
    {
      "field": "uses",
      "target": "09ZJ6CAK7S"
    },
    {
      "field": "uses",
      "target": "04DC0SRN2K"
    },
    {
      "field": "uses",
      "target": "04QEBF8S49"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/MRH07WB74D.md",
  "msc": [
    "20D20"
  ],
  "name": "Proof that the number of Sylow subgroups divides the index",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MRH/MRH07WB74D.md",
  "status": null,
  "tag": "MRH07WB74D",
  "uri": "m2://MRH07WB74D",
  "url": "https://math2.tamnd.com/wiki/proof-that-the-number-of-sylow-subgroups-divides-the-index/",
  "writing": {
    "tag": "MRH07WB74D",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}