{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a prime, and fix a $p$-modular system with residue field $k$ of characteristic $p$.\nThe $p$-blocks of a finite group $H$ are the blocks of the [[group algebra|MNP2HBWB5Q]] $k[H]$.\nEach irreducible complex [[character|RMWMYBY8TC]] and each irreducible [[Brauer character|R2N3GBDBVQ]] of $H$ lies in exactly one block.\nEach block $b$ has a defect group, a [[$p$-subgroup|06QYRTX9KQ]] of $H$ determined up to conjugacy.\nEach block $b$ of $H$ has a central character $\\lambda_b \\colon Z(k[H]) \\to k$.\nFor a subset $X$ of $G$, let $\\hat{X}$ be the sum of the elements of $X$ in $k[G]$.\nLet $H \\le G$ and let $b$ be a block of $H$.\nSuppose there is a block $B$ of $G$ with $\\lambda_B(\\hat{C}) = \\lambda_b(\\widehat{C \\cap H})$ for each [[conjugacy class|0DEBXAJ5SA]] $C$ of $G$.\nThen $B$ is unique, one says that $b^G$ is defined, and $b^G = B$ is the Brauer correspondent of $b$.\nThe principal block is the block containing the trivial character.\n\n1. First main theorem. Let $D$ be a $p$-subgroup of $G$ with [[normaliser|0YGN3VAA0D]] $N_G(D)$. The map $b \\mapsto b^G$ is a bijection from the blocks of $N_G(D)$ with defect group $D$ onto the blocks of $G$ with defect group $D$.\n2. Second main theorem. Let $x \\in G$ have order a power of $p$, and let $H = C_G(x)$ be its [[centraliser|02YYV36BFM]]. Let $\\chi$ be an irreducible complex character of $G$. There are unique complex numbers $d^x_{\\chi\\varphi}$, for $\\varphi$ in the set of irreducible Brauer characters of $H$, such that $\\chi(xy) = \\sum_{\\varphi} d^x_{\\chi\\varphi} \\, \\varphi(y)$ for every $p$-regular $y \\in H$. Let $\\chi$ lie in the block $B$ and $\\varphi$ in the block $b$ of $H$. If $d^x_{\\chi\\varphi} \\ne 0$, then $b^G = B$.\n3. Third main theorem. Let $H \\le G$ and let $b$ be a block of $H$ with defect group $D$ such that $C_G(D) \\le H$. Then $b^G$ is defined, and $b^G$ is the principal block of $G$ if and only if $b$ is the principal block of $H$.\n\n## Remarks\n\nThe numbers $d^x_{\\chi\\varphi}$ are the generalised decomposition numbers; for $x = 1$ they are the ordinary decomposition numbers.\nThe first main theorem reduces the count of blocks with a given defect group to a question about the normaliser of a $p$-subgroup, a local subgroup.\nFor example, if $D$ is a normal Sylow $p$-subgroup of $G$, then $N_G(D) = G$ and the theorem is a tautology; the content appears when $N_G(D)$ is much smaller than $G$.\nThe second main theorem says that the values of characters of a block at elements with $p$-part $x$ are controlled by blocks of $C_G(x)$ that correspond to it.\nThe third main theorem identifies the principal block of $G$ through principal blocks of local subgroups, which is used in the local analysis of finite simple groups.\n\n## Explanations\n\n### L1 human\n\nBrauer's three main theorems relate the blocks of a finite group, the natural pieces of its representation theory modulo $p$, to blocks of small subgroups built from $p$-subgroups.\nThey make precise the idea that much of the $p$-modular representation theory of a group is controlled locally, near its $p$-subgroups.\n\n### L2 human\n\nThese results are the foundation of block theory.\nThe first gives the Brauer correspondence with normalisers, the second computes character values through centralisers of $p$-elements, and the third tracks the principal block.\nTogether they underlie local-global conjectures such as the [[McKay conjecture|WQQTFRZFBD]], Alperin's weight conjecture and [[Brauer's height zero conjecture|5M0HYY3RYV]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "R2N3GBDBVQ",
      "RMWMYBY8TC"
    ],
    "kind": "statement",
    "msc": [
      "20C20"
    ],
    "name": "Brauer's main theorems",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4958221",
      "wikipedia": "Brauer's three main theorems"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Brauer%27s_three_main_theorems"
      }
    ],
    "tag": "NB057FEAAE",
    "uses": [
      "G5S1M9",
      "06QYRTX9KQ",
      "0YGN3VAA0D",
      "02YYV36BFM",
      "0DEBXAJ5SA",
      "RMWMYBY8TC",
      "R2N3GBDBVQ",
      "MNP2HBWB5Q"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/NB057FEAAE/",
  "links": [
    {
      "field": "about",
      "target": "R2N3GBDBVQ"
    },
    {
      "field": "about",
      "target": "RMWMYBY8TC"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "06QYRTX9KQ"
    },
    {
      "field": "uses",
      "target": "0YGN3VAA0D"
    },
    {
      "field": "uses",
      "target": "02YYV36BFM"
    },
    {
      "field": "uses",
      "target": "0DEBXAJ5SA"
    },
    {
      "field": "uses",
      "target": "RMWMYBY8TC"
    },
    {
      "field": "uses",
      "target": "R2N3GBDBVQ"
    },
    {
      "field": "uses",
      "target": "MNP2HBWB5Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NB057FEAAE.md",
  "msc": [
    "20C20"
  ],
  "name": "Brauer's main theorems",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NB0/NB057FEAAE.md",
  "status": null,
  "tag": "NB057FEAAE",
  "uri": "m2://NB057FEAAE",
  "url": "https://math2.tamnd.com/wiki/brauers-main-theorems/",
  "writing": {
    "tag": "NB057FEAAE",
    "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
      }
    ]
  }
}