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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*).

- Introduction
- One dimension
- Two dimensions
- Three dimensions
- Symmetry groups in general
- Group structure in terms of symmetries
- See also
- Further reading
- External links

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 **R**^{n}).

Two geometric figures have the same *symmetry type* when their symmetry groups are * conjugate * subgroups of the Euclidean group: that is, when the subgroups *H*_{1}, *H*_{2} are related by *H*_{1} = *g*^{−1}*H*_{2}*g* for some *g* in E(*n*). For example:

- two 3D figures have mirror symmetry, but with respect to different mirror planes.
- two 3D figures have 3-fold rotational symmetry, but with respect to different axes.
- two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction.

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.

The isometry groups in one dimension are:

- the trivial cyclic group C
_{1} - the groups of two elements generated by a reflection; they are isomorphic with C
_{2} - the infinite discrete groups generated by a translation; they are isomorphic with
**Z**, the additive group of the integers - the infinite discrete groups generated by a translation and a reflection; they are isomorphic with the generalized dihedral group of
**Z**, Dih(**Z**), also denoted by D_{∞}(which is a semidirect product of**Z**and C_{2}). - the group generated by all translations (isomorphic with the additive group of the real numbers
**R**); this group cannot be the symmetry group of a Euclidean figure, even endowed with a pattern: such a pattern would be homogeneous, hence could also be reflected. However, a constant one-dimensional vector field has this symmetry group. - the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group Dih(
**R**).

Up to conjugacy the discrete point groups in two-dimensional space are the following classes:

- cyclic groups C
_{1}, C_{2}, C_{3}, C_{4}, ... where C_{n}consists of all rotations about a fixed point by multiples of the angle 360°/*n* - dihedral groups D
_{1}, D_{2}, D_{3}, D_{4}, ..., where D_{n}(of order 2*n*) consists of the rotations in C_{n}together with reflections in*n*axes that pass through the fixed point.

C_{1} is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C_{2} is the symmetry group of the letter "Z", C_{3} that of a triskelion, C_{4} of a swastika, and C_{5}, C_{6}, etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four.

D_{1} 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".

D_{2}, 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.

D_{3}, D_{4} etc. are the symmetry groups of the regular polygons.

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:

- the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S
^{1}, the multiplicative group of complex numbers of absolute value 1. It is the*proper*symmetry group of a circle and the continuous equivalent of C_{n}. There is no geometric figure that has as*full*symmetry group the circle group, but for a vector field it may apply (see the three-dimensional case below). - the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S
^{1}) as it is the generalized dihedral group of S^{1}.

Non-bounded figures may have isometry groups including translations; these are:

- the 7 frieze groups
- the 17 wallpaper groups
- for each of the symmetry groups in one dimension, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction
- ditto with also reflections in a line in the first direction.

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).

The continuous symmetry groups with a fixed point include those of:

- cylindrical symmetry without a symmetry plane perpendicular to the axis. This applies, for example, to a bottle or cone.
- cylindrical symmetry with a symmetry plane perpendicular to the axis
- spherical symmetry

For objects with scalar field patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for vector field patterns: for example, in cylindrical coordinates with respect to some axis, the vector field has cylindrical symmetry with respect to the axis whenever and have this symmetry (no dependence on ); and it has reflectional symmetry only when .

For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.

The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.

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.

For example, let *G* = Sym(*X*) be the finite symmetry group of a figure *X* in a Euclidean space, and let *H*⊂*G* be a subgroup. Then *H* can be interpreted as the symmetry group of *X*^{+}, a "decorated" version of *X*. Such a decoration may be constructed as follows. Add some patterns such as arrows or colors to *X* so as to break all symmetry, obtaining a figure *X*^{#} with Sym(*X*^{#}) = {1}, the trivial subgroup; that is, *gX*^{#}≠*X*^{#} for all non-trivial *g*∈*G*. Now we get:

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*.

As an example, consider the dihedral group *G* = *D*_{3} = Sym(*X*), where *X* is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure *X*^{#}. Letting τ∈*G* be the reflection of the arrowed edge, the composite figure *X*^{+} = *X*^{#}∪τ*X*^{#} has a bidirectional arrow on that edge, and its symmetry group is *H* = {1, τ}. This subgroup is not normal, since *gX*^{+} may have the bi-arrow on a different edge, giving a different reflection symmetry group.

However, letting H = {1, ρ, ρ^{2}} ⊂*D*_{3} be the cyclic subgroup generated by a rotation, the decorated figure *X*^{+} consists of a 3-cycle of arrows with consistent orientation. Then *H* is normal, since drawing such a cycle with either orientation yields the same symmetry group *H*.

- Burns, G.; Glazer, A. M. (1990).
*Space Groups for Scientists and Engineers*(2nd ed.). Boston: Academic Press, Inc. ISBN 0-12-145761-3. - Clegg, W (1998).
*Crystal Structure Determination (Oxford Chemistry Primer)*. Oxford: Oxford University Press. ISBN 0-19-855901-1. - O'Keeffe, M.; Hyde, B. G. (1996).
*Crystal Structures; I. Patterns and Symmetry*. Washington, DC: Mineralogical Society of America, Monograph Series. ISBN 0-939950-40-5. - Miller, Willard Jr. (1972).
*Symmetry Groups and Their Applications*. New York: Academic Press. OCLC 589081. Archived from the original on 2010-02-17. Retrieved 2009-09-28.

- Weisstein, Eric W. "Symmetry Group".
*MathWorld*. - Weisstein, Eric W. "Tetrahedral Group".
*MathWorld*. - Overview of the 32 crystallographic point groups - form the first parts (apart from skipping
*n*=5) of the 7 infinite series and 5 of the 7 separate 3D point groups

In abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In mathematics, a **dihedral group** is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

Some elementary **examples of groups** in mathematics are given on Group (mathematics). Further examples are listed here.

In Euclidean geometry, a **translation** is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

In mathematics, a **frieze** or **frieze pattern** is a two-dimensional design that repeats in one direction. The term is derived from architecture and decorative arts, where such repeating patterns are often used. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a **frieze group**.

In geometry, a **glide reflection** or **transflection** is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Because the distances between points are not changed under glide reflection, it is a motion or isometry. When the context is the two-dimensional Euclidean plane, the hyperplane of reflection is a straight line called the *glide line* or *glide axis*. When the context is three-dimensional space, the hyperplane of reflection is a plane called the *glide plane*. The displacement vector of the translation is called the *glide vector*.

In mathematics, a topological group *G* is called a **discrete group** if there is no limit point in it. Equivalently, the group *G* is discrete if and only if its identity is isolated.

In mathematics, a **Euclidean group** is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension *n* of the space, and is commonly denoted E(*n*) or ISO(*n*).

**Rotational symmetry**, also known as **radial symmetry** in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

In geometry, a **point group** is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension *d* is then a subgroup of the orthogonal group O(*d*). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.

In geometry, a **Euclidean plane isometry** is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

A **one-dimensional symmetry group** is a mathematical group that describes symmetries in one dimension (1D).

In group theory and geometry, a **reflection group** is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections, it is a continuous group, not a discrete group, and is generally considered separately.

In geometry, a **point group in three dimensions** is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

In geometry, **orbifold notation** is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.

In a group, the **conjugate** by *g* of *h* is *ghg*^{−1}.

In geometry, a **two-dimensional point group** or **rosette group** is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

In geometry, an **apeirogon** or **infinite polygon** is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.

In mathematics, the **generalized dihedral groups** are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group *O*(2). Dihedral groups play an important role in group theory, geometry, and chemistry.

In geometry, an object has **symmetry** if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be *symmetric under rotation* or to have *rotational symmetry*. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

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