{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$, and let crystallographic groups be as in [[crystallographic group|7KTJKFQT2N]].\n\n1. Let $\\Gamma$ be a crystallographic group of dimension $n$ and let $T$ be the subgroup of [[translations|087VJM5Q77]] in $\\Gamma$. Then $T$ is a [[free abelian group|0WJX8MGG37]] of rank $n$ whose elements span $\\R^n$, and $T$ is a [[normal subgroup|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]] in $\\Gamma$.\n2. Let $\\Gamma_1$ and $\\Gamma_2$ be crystallographic groups of dimension $n$. If $\\Gamma_1$ and $\\Gamma_2$ are [[isomorphic|08MG7H8JW5]] as abstract groups, then there is a bijective [[affine map|0JF0TVYGWT]] $\\alpha \\colon \\R^n \\to \\R^n$ with $\\Gamma_2 = \\alpha \\Gamma_1 \\alpha^{-1}$.\n3. For each $n$, there are only finitely many crystallographic groups of dimension $n$ up to isomorphism.\n\n## Remarks\n\nThe proof of part 1 uses a commutator estimate.\nIf two isometries have linear parts close to the identity, their commutator has a linear part closer still.\nDiscreteness then forces enough elements to be pure translations.\nPart 3 follows from part 1, because the finite quotient $\\Gamma / T$ acts faithfully on $T \\cong \\Z^n$.\nThere are finitely many finite subgroups of $\\mathrm{GL}_n(\\Z)$ up to conjugacy, and each gives finitely many extensions.\n\nExample: in dimension $1$ there are two crystallographic groups up to isomorphism.\nThey are the translations $x \\mapsto x + k$ with $k \\in \\Z$, and the infinite dihedral group of all maps $x \\mapsto \\pm x + k$ with $k \\in \\Z$.\nIn both, the translation subgroup is $\\Z$, of index $1$ and $2$.\n\nThe compactness of the quotient is necessary.\nThe translations $(x, y) \\mapsto (x + k, y)$ with $k \\in \\Z$ form a discrete group of isometries of $\\R^2$ whose translations span only a line.\n\n## Explanations\n\n### L1 human\n\nEvery crystal pattern in $n$-dimensional space repeats along $n$ independent directions.\nIts symmetry group is mostly made of translations, with only finitely many kinds of extra rotations and reflections on top.\nTwo such symmetry groups that look the same as abstract groups really are the same, after a change of coordinates.\nAnd in each dimension the list of possible crystal symmetry groups is finite.\n\n### L2 human\n\nPart 3 is the reason why there are exactly $17$ [[wallpaper groups|MY3CBHARYQ]] in the plane and $219$ space groups in dimension $3$ up to affine conjugacy.\nA torsion-free crystallographic group acts freely on $\\R^n$, and its quotient is a compact flat Riemannian manifold.\nSo the theorems say that each compact flat manifold is finitely covered by a torus, and that there are finitely many of them in each dimension up to affine equivalence.\nThe argument behind part 1 is close to the proof of the [[Jordan–Schur theorem|XT6EEXPD4P]] on finite groups of matrices.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "7KTJKFQT2N"
    ],
    "kind": "statement",
    "msc": [
      "20H15",
      "52C22"
    ],
    "name": "Bieberbach theorems",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Space_group"
      }
    ],
    "tag": "YE87R5CDJY",
    "uses": [
      "7KTJKFQT2N",
      "0H3YY8CPKH",
      "087VJM5Q77",
      "0WJX8MGG37",
      "0CB0DX5BVV",
      "0DTXBQPA2S",
      "08MG7H8JW5",
      "0JF0TVYGWT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/YE87R5CDJY/",
  "links": [
    {
      "field": "about",
      "target": "7KTJKFQT2N"
    },
    {
      "field": "uses",
      "target": "7KTJKFQT2N"
    },
    {
      "field": "uses",
      "target": "0H3YY8CPKH"
    },
    {
      "field": "uses",
      "target": "087VJM5Q77"
    },
    {
      "field": "uses",
      "target": "0WJX8MGG37"
    },
    {
      "field": "uses",
      "target": "0CB0DX5BVV"
    },
    {
      "field": "uses",
      "target": "0DTXBQPA2S"
    },
    {
      "field": "uses",
      "target": "08MG7H8JW5"
    },
    {
      "field": "uses",
      "target": "0JF0TVYGWT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YE87R5CDJY.md",
  "msc": [
    "20H15",
    "52C22"
  ],
  "name": "Bieberbach theorems",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YE8/YE87R5CDJY.md",
  "status": null,
  "tag": "YE87R5CDJY",
  "uri": "m2://YE87R5CDJY",
  "url": "https://math2.tamnd.com/wiki/bieberbach-theorems/",
  "writing": {
    "tag": "YE87R5CDJY",
    "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
      }
    ]
  }
}