In

^{''n''}).
Two geometric figures have the same ''symmetry type'' when their symmetry groups are ''_{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

_{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 _{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).
See also

_{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''
*_{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 _{2} is the symmetry group of the letter "Z", C_{3} that of a _{4} of a _{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 _{2}, which is isomorphic to the _{3}, D_{4} etc. are the symmetry groups of the ^{1}, the multiplicative group of _{''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

^{+}, 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:
:$X^+\; \backslash \; =\; \backslash \; \backslash bigcup\_\; hX^\; \backslash quad\backslash text\backslash quad\; H\; =\; \backslash mathrm(X^+).$
^{+} is the conjugate subgroup ''gHg''^{−1}. Thus ''H'' is normal whenever:
:$\backslash mathrm(gX^+)\; =\; \backslash mathrm(X^+)\; \backslash \; \backslash \; \backslash text\; \backslash \; g\backslash in\; G;$
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''.

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 Geometry Symmetry Group theory

group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, the symmetry group of a geometric object is the group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

of all transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

s under which the object is invariant, endowed with the group operation of composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

. Such a transformation is an invertible mapping of the ambient space
An ambient space or ambient configuration space is the space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, al ...

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
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

space, its symmetries form a subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

of the isometry group In , the isometry group of a is the of all (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the as operation. Its is the . The elements of the isometry group are sometimes called s of the space.
Every isomet ...

of the ambient space. This article mainly considers symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

groups in Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...

, 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 ascalar field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, a function of position with values in a set of colors or substances; as a vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

; or as a more general function on the object.) The group of isometries of space induces a group action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 reflection
In 2-dimensional geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propert ...

s and improper rotation
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

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
Chirality is a property of important in several branches of science. The word ''chirality'' is derived from the (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from its ; that is, i ...

when it has no orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building design ...

-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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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
In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...

of the group (the images of a given point under all group elements) forms a discrete set
Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual.
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic c ...

. All finite symmetry groups are discrete.
Discrete symmetry groups come in three types: (1) finite point group
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

s, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s containing elements of both previous types, and perhaps also extra transformations like screw displacement
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotat ...

s and glide reflections. There are also continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...

groups (Lie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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 Rconjugate
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Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the change ...

'' subgroups of the Euclidean group: that is, when the subgroups ''H''rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related ...

, 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
Celestial mechanics is the branch of astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical obje ...

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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

; 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 trivialcyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

Csemidirect product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of Z and Csymmetry groups in one dimension
A one-dimensional symmetry group is a group (mathematics), mathematical group that describes symmetry, symmetries in one dimension (1D).
A pattern in 1D can be represented as a function ''f''(''x'') for, say, the color at position ''x''.
The on ...

.
Two dimensions

Up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

conjugacy the discrete point groups in two-dimensional space are the following classes:
*cyclic groups Cdihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

s Dtrivial groupIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". Ctriskelion
A triskelion or triskeles is a motif consisting of a triple spiral exhibiting rotational symmetry.
The spiral design can be based on interlocking Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral n ...

, Cswastika
The swastika symbol, 卐 (''right-facing'' or ''clockwise'') or 卍 (''left-facing'', ''counterclockwise'', or sauwastika), is an ancient religious icon
An icon (from the Greek language, Greek 'image, resemblance') is a religious work ...

, and Cbilateral 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 pla ...

, for example the letter "A".
DKlein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...

, 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.
Dregular polygon
In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the ...

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 system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

Scomplex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s of absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

1. It is the ''proper'' symmetry group of a circle and the continuous equivalent of Cfrieze groupIn mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction. Such patterns occur frequently in architecture and decorative art. A frieze group is the set of symmetry, symmetries of a frieze ...

s
*the 17 wallpaper groupImage:Wallpaper group-p4m-5.jpg, 250px, Example of an Egyptian design with wallpaper group #Group p4mm, ''p''4''m''
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional rep ...

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 Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "cold drop, frozen drop" ...

, 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 group
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "col ...

s (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 (clay, glass, plastic, aluminium etc.) in various shapes and sizes to store and transport liquids (water, milk, beer, wine, ink, cooking oil, medicine, soft drinks, shampoo ...

*cylindrical symmetry with a symmetry plane perpendicular to the axis
*spherical symmetry
For objects with scalar field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

patterns: for example, in cylindrical coordinates
240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height .
A cylindrical coordinate system is a three-dimensional coordinate system that s ...

with respect to some axis, the vector field
$\backslash mathbf\; =\; A\_\backslash rho\backslash boldsymbol\; +\; A\_\backslash phi\backslash boldsymbol\; +\; A\_z\backslash boldsymbol$ has cylindrical symmetry with respect to the axis whenever $A\_\backslash rho,\; A\_\backslash phi,$ and $A\_z$ have this symmetry (no dependence on $\backslash phi$); and it has reflectional symmetry only when $A\_\backslash phi\; =\; 0$.
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
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rot ...

, such as an infinite helix
A helix (), plural helixes or helices (), is a shape like a corkscrew or spiral staircase. It is a type of smooth
Smooth may refer to:
Mathematics
* Smooth function
is a smooth function with compact support.
In mathematical analysis, the ...

. See also subgroups of the Euclidean group.
Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, orautomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

group. Each type of mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

has 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
In mathematics, the Erlangen program is a method of characterizing geometry, geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' ...

.
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 geome ...

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 (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...

is the symmetry group of its Cayley graph on two generators ''a'' and ''b''
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

; the free group
for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the stud ...

is the symmetry group of an infinite tree graph
In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only wood plants with secondary ...

.
Group structure in terms of symmetries

Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group (mathematics), group ''G'' is group isomorphism, isomorphic to a subgroup of the symmetric group acting on ''G''. This can be understood as an example of ...

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''Normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s may also be characterized in this framework.
The symmetry group of the translation ''gX'' See also

Further reading

* * * *External links

* * {{MathWorld , urlname=TetrahedralGroup , title=Tetrahedral GroupOverview 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 Geometry Symmetry Group theory