International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 529530
Section 3.4.4.2. Layer groups^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
An adequate concept for characterizing symmetry properties of simple domain twins and planar domain walls is that of layer groups. A layer group describes the symmetry of objects that exist in a threedimensional space and have twodimensional translation symmetry. Typical examples are twodimensional planes in threedimensional space [twosided planes and sectional layer groups (Holser, 1958a,b), domain walls and interfaces of zero thickness], layers of finite thickness (e.g. domain walls and interfaces of finite thickness) and two semiinfinite crystals joined along a planar and coherent (compatible) interface [e.g. simple domain twins with a compatible (coherent) domain wall, bicrystals].
A crystallographic layer group comprises symmetry operations (isometries) that leave invariant a chosen crystallographic plane p in a crystalline object. There are two types of such operations:
A layer group consists of two parts: where is a subgroup of that comprises all sidepreserving operations of ; this group is isomorphic to a plane group and is called a trivial layer group or a face group. An underlined character denotes a sidereversing operation and the left coset contains all sidereversing operations of . Since is a halving subgroup, the layer group can be treated as a dichromatic (blackandwhite) group in which sidepreserving operations are colourpreserving operations and sidereversing operations are colourexchanging operations.
There are 80 layer groups with discrete twodimensional translation subgroups [for a detailed treatment see IT E (2010), or e.g. Vainshtein (1994), Shubnikov & Kopcik (1974), Holser (1958a)]. Equivalent names for these layer groups are net groups (Opechowski, 1986), plane groups in three dimensions (Grell et al., 1989), groups in a twosided plane (Holser, 1958a,b) and others.
To these layer groups there correspond 31 point groups that describe the symmetries of crystallographic objects with twodimensional continuous translations. Holser (1958b) calls these groups point groups in a twosided plane, Kopský (1993) coins the term pointlike layer groups. We shall use the term `layer groups' both for layer groups with discrete translations, used in a microscopic description, and for crystallographic `pointlike layer groups' with continuous translations in the continuum approach. The geometrical meaning of these groups is similar and most of the statements and formulae hold for both types of layer groups.
Crystallographic layer groups with a continuous translation group [point groups of twosided plane (Holser, 1958b)] are listed in Table 3.4.4.1. The international notation corresponds to international symbols of layer groups with discrete translations; this notation is based on the Hermann–Mauguin (international) symbols of threedimensional space groups, where the c direction is the direction of missing translations and the character `1' represents a symmetry direction in the plane with no associated symmetry element (see IT E , 2010).

In the noncoordinate notation (Janovec, 1981), sidereversing operations are underlined. Thus e.g. denotes a rotation around a twofold axis in the plane p and a reflection through this plane, whereas 2 is a sidepreserving rotation around an axis perpendicular to the plane and m is a sidepreserving reflection through a plane perpendicular to the plane p. With exception of and , the symbol of an operation specifies the orientation of the plane p. This notation allows one to signify layer groups with different orientations in one reference coordinate system. Another noncoordinate notation has been introduced by Shubnikov & Kopcik (1974).
If a crystal with pointgroup symmetry G is bisected by a crystallographic plane p, then all operations of G that leave the plane p invariant form a sectional layer group = of the plane p in G. Operations of the group can be divided into two sets [see equation (3.4.4.7)]: where the trivial layer group expresses the symmetry of the crystal face with normal . These face symmetries are listed in IT A (2005), Part 10 , for all crystallographic point groups G and all orientations of the plane expressed by Miller indices . The underlined operation is a sidereversing operation that inverts the normal . The left coset contains all sidereversing operations of .
The number of planes symmetryequivalent (in G) with the plane p is equal to the index of in G:
Example 3.4.4.1. As an example, we find the sectional layer group of the plane in the group (see Fig. 3.4.2.2).
In this example and the plane crystallographically equivalent with the plane is the plane (100) with sectional symmetry .
References
International Tables for Crystallography (2005). Vol. A, SpaceGroup Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.International Tables for Crystallography (2010). Vol. E, Subperiodic groups, 2nd. ed., edited by V. Kopský & D. B. Litvin. Chichester: Wiley.
Grell, H., Krause, C. & Grell, J. (1989). Tables of the 80 Plane Space Groups in Three Dimensions. Berlin: Akademie der Wissenschaften der DDR.
Holser, W. T. (1958a). Relation of symmetry to structure in twinning. Z. Kristallogr. 110, 249–265.
Holser, W. T. (1958b). Point groups and plane groups in a twosided plane and their subgroups. Z. Kristallogr. 110, 266–281.
Janovec, V. (1981). Symmetry and structure of domain walls. Ferroelectrics, 35, 105–110.
Kopský, V. (1993). Translation normalizers of Euclidean motion groups. J. Math. Phys. 34, 1548–1576.
Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: NorthHolland.
Shubnikov, A. V. & Kopcik, V. A. (1974). Symmetry in Science and Art. New York: Plenum Press.
Vainshtein, B. K. (1994). Modern Crystallography I. Symmetry of Crystals. Berlin: Springer.