Math 2.0

Group theory and generalizations Linear and matrix groups Theorem

Bieberbach theorems

Builds on Crystallographic groupEuclidean groupTranslationFree abelian groupNormal subgroupIndex of a subgroupGroup isomorphismAffine map

Statement

Let n≥1n \ge 1, and let crystallographic groups be as in crystallographic group.

  1. Let Γ\Gamma be a crystallographic group of dimension nn and let TT be the subgroup of translations in Γ\Gamma. Then TT is a free abelian group of rank nn whose elements span ℝn\R^n, and TT is a normal subgroup of finite index in Γ\Gamma.
  2. Let Γ1\Gamma_1 and Γ2\Gamma_2 be crystallographic groups of dimension nn. If Γ1\Gamma_1 and Γ2\Gamma_2 are isomorphic as abstract groups, then there is a bijective affine map α ⁣:ℝn→ℝn\alpha \colon \R^n \to \R^n with Γ2=αΓ1α−1\Gamma_2 = \alpha \Gamma_1 \alpha^{-1}.
  3. For each nn, there are only finitely many crystallographic groups of dimension nn up to isomorphism.

Remarks

The proof of part 1 uses a commutator estimate. If two isometries have linear parts close to the identity, their commutator has a linear part closer still. Discreteness then forces enough elements to be pure translations. Part 3 follows from part 1, because the finite quotient Γ/T\Gamma / T acts faithfully on T≅ℤnT \cong \Z^n. There are finitely many finite subgroups of GLn(ℤ)\mathrm{GL}_n(\Z) up to conjugacy, and each gives finitely many extensions.

Example: in dimension 11 there are two crystallographic groups up to isomorphism. They are the translations x↦x+kx \mapsto x + k with k∈ℤk \in \Z, and the infinite dihedral group of all maps x↦±x+kx \mapsto \pm x + k with k∈ℤk \in \Z. In both, the translation subgroup is ℤ\Z, of index 11 and 22.

The compactness of the quotient is necessary. The translations (x,y)↦(x+k,y)(x, y) \mapsto (x + k, y) with k∈ℤk \in \Z form a discrete group of isometries of ℝ2\R^2 whose translations span only a line.

Going deeper

For a student

Part 3 is the reason why there are exactly 1717 wallpaper groups in the plane and 219219 space groups in dimension 33 up to affine conjugacy. A torsion-free crystallographic group acts freely on ℝn\R^n, and its quotient is a compact flat Riemannian manifold. So 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. The argument behind part 1 is close to the proof of the Jordan–Schur theorem on finite groups of matrices.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Affine map, concept, no evidence levelAffine mapconceptCrystallographic group, concept, no evidence levelCrystallographic groupconceptEuclidean isometry, concept, no evidence levelEuclidean isometryconceptFree abelian group, concept, no evidence levelFree abelian groupconceptGroup isomorphism, concept, no evidence levelGroup isomorphismconceptIndex of a subgroup, concept, no evidence levelIndex of a subgroupconceptNormal subgroup, concept, no evidence levelNormal subgroupconceptTranslation, concept, no evidence levelTranslationconceptEuclidean group, object, no evidence levelEuclidean groupobjectBieberbach theorems, statement, no evidence levelBieberbach theoremsstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Affine map concept
  2. Crystallographic group concept
  3. Euclidean isometry concept
  4. Free abelian group concept
  5. Group isomorphism concept
  6. Index of a subgroup concept
  7. Normal subgroup concept
  8. Translation concept
  9. Euclidean group object
  10. Bieberbach theorems statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Space_group
About this page
TagYE87R5CDJY
URIm2://YE87R5CDJY
Kindstatement, theorem
WritingW0 stub
MSC20H15, 52C22
StatusThis copy of the site has no build data. Run m2 status YE87R5CDJY.
AboutCrystallographic group
UsesCrystallographic group, Euclidean group, Translation, Free abelian group, Normal subgroup, Index of a subgroup, Group isomorphism, Affine map
Changed2 November 2026

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