The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation. It is a subgroup of the isometry group of the space concerned.
(If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts; see below.)
The "objects" may be geometric figures, images and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g. a function of position with values in a set of colors. For symmetry of e.g. 3D bodies one may also want to take physical composition into account. The group of isometries of space induces a group action on objects in it.
The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations and compositions of these) which leave the figure invariant is called its proper symmetry group. The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant).
Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is a subgroup of the special orthogonal group SO(n) then, and therefore also called rotation group of the figure.
Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversion and rotoinversion - they are in fact just the finite subgroups of O(n), (2) infinite lattice groups, which include only translations, and (3) infinite space groups which combines elements of both previous types, and may also include extra transformations like screw axis and glide reflection. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups.
Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(n) (the isometry group of Rn), where two subgroups H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1=g-1H2g. For example:
Sometimes a broader concept of "same symmetry type" is used, resulting in e.g. 17 wallpaper groups.
When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This excludes for example in 1D the group of translations by a rational number. A "figure" with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous.
See also symmetry groups in one dimension.
Up to conjugacy the discrete point groups in 2 dimensional space are the following classes:
C1 is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, 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, and D3, D4 etc. are the symmetry groups of the regular polygons.
The actual symmetry groups in each of these cases have 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 2D with a fixed point, where for all points the set of images under the isometries is topologically closed are:
For non-bounded figures, the additional isometry groups can include translations; the closed ones are:
The continuous symmetry groups with a fixed point include those of:
For objects and scalar fields the cylindrical symmetry implies vertical planes of reflection. However, for vector fields it does not: in cylindrical coordinates with respect to some axis, has cylindrical symmetry with respect to the axis if and only if and have this symmetry, i.e., they do not depend on φ. Additionally there is reflectional symmetry if and only if .
For spherical symmetry there is no such distinction, it implies planes of reflection.
For example, automorphism groups of certain models of finite geometries are not "symmetry groups" in the usual sense, although they preserve symmetry. They do this by preserving families of point-sets rather than point-sets (or "objects") themselves.
Like above, the group of automorphisms of space induces a group action on objects in it.
For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same (here "the same" does not mean something like e.g. "the same up to translation and rotation", but it means "exactly the same"). Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure.
There is a bijection between every pair of equivalence classes: the inverse of a representative of the first equivalence class, composed with a representative of the second.
In the case of a finite automorphism group of the whole space, its order is the order of the symmetry group of the figure multiplied by the number of isomorphic versions of the figure.
Compare Lagrange's theorem (group theory) and its proof.