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In
group theory 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 ...
, 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 In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is t ...
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
groups in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, but the concept may also be studied for more general types of geometric structure.

# Introduction

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 In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ...
, 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 In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
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 rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
s), 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 In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
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 In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
. That is, every
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of the group (the images of a given point under all group elements) forms a
discrete set ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
. 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 group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unc ...
s containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections. There are also continuous symmetry groups (
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
s), 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 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 In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
are topologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.

# One dimension

The isometry groups in one dimension are: *the trivial
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
C1 *the groups of two elements generated by a reflection; they are isomorphic with C2 *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 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). Dihe ...
of Z, Dih(Z), also denoted by D (which is a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
of Z and C2). *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).

# Two dimensions

Up to conjugacy the discrete point groups in two-dimensional space are the following classes: *cyclic groups C1, C2, C3, C4, ... where C''n'' consists of all rotations about a fixed point by multiples of the angle 360°/''n'' *
dihedral group 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, g ...
s D1, D2, D3, D4, ..., 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. C1 is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
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 A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of ...
, C4 of a
swastika The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by the Nazi Party and by neo-Nazis. ...
, 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 Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a pl ...
, for example the letter "A". D2, which is isomorphic to the
Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...
, 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. D3, D4 etc. are the symmetry groups of the
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s. Within each of these symmetry types, there are two
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
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 In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
S1, the multiplicative group of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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(S1) as it is the generalized dihedral group of S1. Non-bounded figures may have isometry groups including translations; these are: *the 7 frieze groups *the 17
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
s *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.

# Three dimensions

Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, 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 for a beer
bottle A bottle is a narrow-necked container made of an impermeable material (such as glass, plastic or aluminium) in various shapes and sizes that stores and transports liquids. Its mouth, at the bottling line, can be sealed with an internal stop ...
*cylindrical symmetry with a symmetry plane perpendicular to the axis *spherical symmetry For objects with
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ...
patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for vector field patterns: for example, in
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
with respect to some axis, the vector field $\mathbf = A_\rho\boldsymbol + A_\phi\boldsymbol + A_z\boldsymbol$ has cylindrical symmetry with respect to the axis whenever $A_\rho, A_\phi,$ and $A_z$ have this symmetry (no dependence on $\phi$); and it has reflectional symmetry only when $A_\phi = 0$. 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 A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices ...

# Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
group. Each type of
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
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 In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
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 In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
is the symmetry group of its Cayley graph; the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1 ...
is the symmetry group of an infinite
tree graph In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by '' ...
.

# Group structure in terms of symmetries

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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, 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''#) = , the trivial subgroup; that is, ''gX''# ≠ ''X''# for all non-trivial ''g'' ∈ ''G''. Now we get: :$X^+ \ = \ \bigcup_ hX^ \quad\text\quad H = \mathrm\left(X^+\right).$
Normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
s 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: :$\mathrm\left(gX^+\right) = \mathrm\left(X^+\right) \ \ \text \ g\in G;$ 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'' = . 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 = ⊂ ''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''.