In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).
For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.
We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, a function of position with values in a set of colors or substances; as a vector field; or as a more general function on the object.) The group of isometries of space induces a group action on objects in it, and the symmetry group Sym(X) consists of those isometries which map X to itself (as well as mapping any further pattern to itself). We say X is invariant under such a mapping, and the mapping is a symmetry of X.
The above is sometimes called the full symmetry group of X to emphasize that it includes orientation-reversing isometries (reflections, glide reflections and improper rotations), as long as those isometries map this particular X to itself. The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group. An object is chiral when it has no orientation-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.
Any symmetry group whose elements have a common fixed point, which is true if the group is finite or the figure is bounded, can be represented as a subgroup of the orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is called the rotation group of the figure.
In a discrete symmetry group, the points symmetric to a given point do not accumulate toward a limit point. That is, every orbit of the group (the images of a given point under all group elements) forms a discrete set. All finite symmetry groups are discrete.
Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite space groups containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections. There are also continuous symmetry groups (Lie groups), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is O(3), the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(n) (the isometry group of Rn).
Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H1, H2 are related by H1 = g−1H2g for some g in E(n). For example:
In the following sections, we only consider isometry groups whose orbits are topologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a rational number; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.
Up to conjugacy the discrete point groups in two-dimensional space are the following classes:
C1 is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C2 is the symmetry group of the letter "Z", C3 that of a triskelion, C4 of a swastika, and C5, C6, etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four.
D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter "A".
D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes.
Within each of these symmetry types, there are two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.
The remaining isometry groups in two dimensions with a fixed point are:
Non-bounded figures may have isometry groups including translations; these are:
Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In crystallography, only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 individual groups from the 7 series, and 5 of the 7 other individuals).
For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.
In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Each type of mathematical structure has invertible mappings which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the Erlangen programme.
For example, objects in a hyperbolic non-Euclidean geometry have Fuchsian symmetry groups, which are the discrete subgroups of the isometry group of the hyperbolic plane, preserving hyperbolic rather than Euclidean distance. (Some are depicted in drawings of Escher.) Similarly, automorphism groups of finite geometries preserve families of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Just as for Euclidean figures, objects in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space.
Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph; the free group is the symmetry group of an infinite tree graph.
Cayley's theorem states that any abstract group is a subgroup of the permutations of some set X, and so can be considered as the symmetry group of X with some extra structure. In addition, many abstract features of the group (defined purely in terms of the group operation) can be interpreted in terms of symmetries.
Normal subgroups may also be characterized in this framework. The symmetry group of the translation gX + is the conjugate subgroup gHg−1. Thus H is normal whenever:
that is, whenever the decoration of X+ may be drawn in any orientation, with respect to any side or feature of X, and still yield the same symmetry group gHg−1 = H.