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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime.\nLet $B$ be a $p$-block of $G$, that is, a block of the [[group algebra|MNP2HBWB5Q]] of $G$ over an algebraically closed field of characteristic $p$.\nLet $D$ be a defect group of $B$.\nWrite $\\abs{G}_p = p^a$ and $\\abs{D} = p^d$.\nEach complex irreducible [[character|RMWMYBY8TC]] $\\chi$ in $B$ has degree $\\chi(1)$ with $p$-part $\\chi(1)_p = p^{a - d + h}$ for an integer $h \\ge 0$, the height of $\\chi$.\nThen every irreducible character in $B$ has height $0$ if and only if $D$ is abelian.\n\n## Remarks\n\nRichard Brauer posed the conjecture in 1955.\nFor the principal block, a defect group is a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$.\nThe direction \"if $D$ is abelian, then all heights are zero\" was proved by Kessar and Malle in 2013, using the [[classification of finite simple groups|06J6P03MG7]].\nThe converse direction was proved by Malle, Navarro, Schaeffer Fry and Tiep, so the conjecture is now a theorem.\nFor $p$-solvable groups the conjecture was proved earlier by Gluck and Wolf.\n\n## Explanations\n\n### L1 human\n\nA block is a piece of the representation theory of a finite group that belongs together when one works modulo a prime $p$.\nEach block has a defect group, a $p$-subgroup that measures how complicated the block is.\nThe conjecture says that a simple numerical pattern in the character degrees of the block, all heights being zero, happens exactly when the defect group is commutative.\nSo local information, a small subgroup, is read off from global information, the character degrees.\n",
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    "history": {
      "first_stated": "1955",
      "posed_by": "Richard Brauer"
    },
    "kind": "conjecture",
    "msc": [
      "20C20",
      "20C15"
    ],
    "name": "Brauer's height zero conjecture",
    "same_as": {
      "wikidata": "Q131191617",
      "wikipedia": "Brauer's height zero conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Brauer%27s_height_zero_conjecture"
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    "tag": "5M0HYY3RYV",
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      "RMWMYBY8TC",
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      "MNP2HBWB5Q"
    ]
  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/5M0HYY3RYV/",
  "links": [
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  "markdown": "https://math2.tamnd.com/node/5M0HYY3RYV.md",
  "msc": [
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    "20C15"
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  "name": "Brauer's height zero conjecture",
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  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5M0/5M0HYY3RYV.md",
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  "tag": "5M0HYY3RYV",
  "uri": "m2://5M0HYY3RYV",
  "url": "https://math2.tamnd.com/wiki/brauers-height-zero-conjecture/",
  "writing": {
    "tag": "5M0HYY3RYV",
    "kind": "conjecture",
    "level": "W0",
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