{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe let a $p$-subgroup act on the cosets of a Sylow subgroup and count fixed points modulo $p$.\n\n1. Write $|G| = p^n m$ with $n \\geq 0$ and $p \\nmid m$. Each [[Sylow $p$-subgroup|0JWZX6FKSH]] of $G$ has order $p^n$.\n2. Let $H$ be a [[$p$-subgroup|06DAH3VS6M]] of $G$, and let $P$ be a Sylow $p$-subgroup of $G$. Such a $P$ exists by the [[first Sylow theorem|0V98WY94B8]].\n3. Let $X = G/P$ be the set of left cosets of $P$. The group $G$ acts on $X$ by $g \\cdot aP = gaP$, by the [[action on left cosets|0AZG100DKH]]. Restrict this action to $H$.\n4. By the [[index formula|04DC0SRN2K]], $|X| = |G|/|P| = p^n m / p^n = m$.\n5. Let $X^H$ be the set of cosets fixed by every element of $H$. Since $H$ is a finite $p$-group, the [[fixed point lemma|098D2E7DEH]] gives $|X^H| \\equiv |X| = m \\pmod p$.\n6. Since $p \\nmid m$, we get $|X^H| \\not\\equiv 0 \\pmod p$. So $X^H$ is not empty. Pick a coset $gP \\in X^H$.\n7. For each $h \\in H$ we have $hgP = gP$. So $g^{-1}hg \\in P$, that is, $h \\in gPg^{-1}$. Hence $H \\subseteq gPg^{-1}$.\n8. The map $x \\mapsto gxg^{-1}$ is a bijection from $P$ onto the [[conjugate subgroup|0D6WHTTH7T]] $gPg^{-1}$. So $gPg^{-1}$ is a subgroup of order $p^n$, a Sylow $p$-subgroup of $G$.\n9. By steps 7 and 8, each $p$-subgroup $H$ lies in some Sylow $p$-subgroup. This is the second claim.\n10. Now let $P$ and $Q$ be Sylow $p$-subgroups. A Sylow $p$-subgroup has order $p^n$, so the [[order of each of its elements|08V7JHS3QZ]] divides $p^n$ and is a power of $p$. So $Q$ is a $p$-subgroup.\n11. Apply steps 3 to 8 with $H = Q$. This gives $g \\in G$ with $Q \\subseteq gPg^{-1}$.\n12. Both $Q$ and $gPg^{-1}$ have $p^n$ elements, and the first is contained in the second. So $Q = gPg^{-1}$. This is the first claim.\n\n## Idea\n\nThe number of cosets of a Sylow subgroup $P$ is prime to $p$. A $p$-group acting on a set fixes a number of points congruent to the size of the set modulo $p$. So some coset $gP$ is fixed, and this forces the acting group inside the conjugate $gPg^{-1}$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "20D20"
    ],
    "name": "Proof of the second Sylow theorem",
    "proves": "09GCQ5STVJ",
    "sources": [
      {
        "locator": "Proof of the Sylow theorems",
        "url": "https://en.wikipedia.org/wiki/Sylow_theorems"
      }
    ],
    "tag": "6M7M12G10R",
    "uses": [
      "0JWZX6FKSH",
      "06DAH3VS6M",
      "0V98WY94B8",
      "0AZG100DKH",
      "098D2E7DEH",
      "04DC0SRN2K",
      "0D6WHTTH7T",
      "08V7JHS3QZ"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/6M7M12G10R/",
  "links": [
    {
      "field": "proves",
      "target": "09GCQ5STVJ"
    },
    {
      "field": "uses",
      "target": "0JWZX6FKSH"
    },
    {
      "field": "uses",
      "target": "06DAH3VS6M"
    },
    {
      "field": "uses",
      "target": "0V98WY94B8"
    },
    {
      "field": "uses",
      "target": "0AZG100DKH"
    },
    {
      "field": "uses",
      "target": "098D2E7DEH"
    },
    {
      "field": "uses",
      "target": "04DC0SRN2K"
    },
    {
      "field": "uses",
      "target": "0D6WHTTH7T"
    },
    {
      "field": "uses",
      "target": "08V7JHS3QZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/6M7M12G10R.md",
  "msc": [
    "20D20"
  ],
  "name": "Proof of the second Sylow theorem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6M7/6M7M12G10R.md",
  "status": null,
  "tag": "6M7M12G10R",
  "uri": "m2://6M7M12G10R",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-second-sylow-theorem/",
  "writing": {
    "tag": "6M7M12G10R",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}