International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 68

The description of a crystal given so far has dealt only with its invariance under the action of the (discrete Abelian) group of translations by vectors of its period lattice Λ.
Let the crystal now be embedded in Euclidean 3space, so that it may be acted upon by the group of rigid (i.e. distancepreserving) motions of that space. The group contains a normal subgroup of translations, and the quotient group may be identified with the 3dimensional orthogonal group . The period lattice Λ of a crystal is a discrete uniform subgroup of .
The possible invariance properties of a crystal under the action of are captured by the following definition: a crystallographic group is a subgroup Γ of if
The two properties are not independent: by a theorem of Bieberbach (1911), they follow from the assumption that Λ is a discrete subgroup of which operates without accumulation point and with a compact fundamental domain (see Auslander, 1965). These two assumptions imply that G acts on Λ through an integral representation, and this observation leads to a complete enumeration of all distinct Γ's. The mathematical theory of these groups is still an active research topic (see, for instance, Farkas, 1981), and has applications to Riemannian geometry (Wolf, 1967).
This classification of crystallographic groups is described elsewhere in these Tables (Wondratschek, 2005), but it will be surveyed briefly in Section 1.3.4.2.2.3 for the purpose of establishing further terminology and notation, after recalling basic notions and results concerning groups and group actions in Section 1.3.4.2.2.2.
References
Auslander, L. (1965). An account of the theory of crystallographic groups. Proc. Am. Math. Soc. 16, 1230–1236.Bieberbach, L. (1911). Über die Bewegungsgruppen der Euklidischen Raume I. Math. Ann. 70, 297–336.
Farkas, D. R. (1981). Crystallographic groups and their mathematics. Rocky Mountain J. Math. 11, 511–551.
Wolf, J. A. (1967). Spaces of Constant Curvature. New York: McGrawHill.
Wondratschek, H. (2005). Introduction to spacegroup symmetry. In International Tables for Crystallography, Vol. A. SpaceGroup Symmetry, edited by Th. Hahn, Part 8. Heidelberg: Springer.