View source for Symmetry group

{{Distinguish|Symmetric group}}
{{About|symmetry groups of geometric objects}}
{{Short description|Group of transformations under which the object is invariant}}
{{no footnotes|date=December 2017}}

[[File:Tetrahedral group 2.svg|thumb|right|300px|A regular [[tetrahedron]] is invariant under twelve distinct [[rotation]]s (if the identity transformation is included as a trivial rotation and reflections are excluded). These are illustrated here in the [[Cycle graph (algebra)|cycle graph]] format, along with the 180° edge (blue arrows) and 120° vertex (pink and orange arrows) rotations that [[permutation|permute]] the tetrahedron through the positions. The twelve rotations form the '''rotation (symmetry) group''' of the figure.]]

In [[group theory]], the '''symmetry group''' of a geometric object is the [[group (mathematics)|group]] of all [[Transformation (geometry)|transformation]]s under which the object is [[invariant (mathematics)|invariant]], endowed with the group operation of [[Function composition|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 (mathematics)|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.

==Introduction==
We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a [[wallpaper group|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 (mathematics)|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 reflection]]s and [[improper rotation]]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 [[chirality (mathematics)|chiral]] when it has no [[Orientation (vector space)|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 (mathematics)|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 group|discrete symmetry group]]''', the points symmetric to a given point do not accumulate toward a [[limit point]]. That is, every [[Orbit (group theory)|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 group]]s''', which include only rotations, reflections, inversions and [[Improper rotation|rotoinversions]] – i.e., the finite subgroups of O(''n''); (2) infinite '''[[Lattice (group)|lattice]] groups''', which include only translations; and (3) infinite '''[[space group]]s''' containing elements of both previous types, and perhaps also extra transformations like [[screw displacement]]s and glide reflections. There are also [[continuous symmetry]] groups ([[Lie group]]s), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is [[orthogonal group|O(3)]], the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the [[Euclidean group#Subgroups|subgroups of the Euclidean group]] E(''n'') (the isometry group of '''R'''<sup>''n''</sup>).

Two geometric figures have the same ''symmetry type'' when their symmetry groups are ''[[Conjugacy class#Conjugacy of subgroups and general subsets|conjugate]]'' subgroups of the Euclidean group: that is, when the subgroups ''H''<sub>1</sub>, ''H''<sub>2</sub> are related by {{nowrap|1=''H''<sub>1</sub> = ''g''<sup>−1</sup>''H''<sub>2</sub>''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 [[Orbit (group theory)|orbits]] are [[Closed (topology)|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.

==One dimension==
{{main|One-dimensional symmetry group}}
The isometry groups in one dimension are:

*the trivial [[cyclic group]] C<sub>1</sub>
*the groups of two elements generated by a reflection; they are isomorphic with C<sub>2</sub>
*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 [[Dihedral group#Generalizations|generalized dihedral group]] of '''Z''',  Dih('''Z'''), also denoted by D<sub>∞</sub> (which is a [[semidirect product]] of '''Z''' and C<sub>2</sub>).
*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 C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, C<sub>4</sub>, ... where C<sub>''n''</sub> consists of all rotations about a fixed point by multiples of the angle 360°/''n''
*[[dihedral group]]s D<sub>1</sub>, D<sub>2</sub>, [[Dihedral group of order 6|D<sub>3</sub>]], [[Examples of groups#The symmetry group of a square: dihedral group of order 8|D<sub>4</sub>]], ..., where D<sub>''n''</sub> (of order 2''n'') consists of the rotations in C<sub>''n''</sub> together with reflections in ''n'' axes that pass through the fixed point.

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

D<sub>1</sub> is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of [[reflection symmetry|bilateral symmetry]], for example the letter "A".

D<sub>2</sub>, 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<sub>3</sub>, D<sub>4</sub> etc. are the symmetry groups of the [[regular polygon]]s.

Within each of these symmetry types, there are two [[degrees of freedom (physics and chemistry)|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<sup>1</sup>, the multiplicative group of [[complex number]]s of [[absolute value]] 1. It is the ''proper'' symmetry group of a circle and the continuous equivalent of C<sub>''n''</sub>. 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<sup>1</sup>) as it is the generalized dihedral group of S<sup>1</sup>.

Non-bounded figures may have isometry groups including translations; these are:
*the 7 [[frieze group]]s 
*the 17 [[wallpaper group]]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==
<!-- This section is linked from [[Bipyramid]] -->
{{see also|Point groups in three dimensions}}

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 theorem|crystallographic restriction]] of the infinite families of general point groups results in 32 [[crystallographic point group]]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, 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 
<math>\mathbf{A} = A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}</math> has cylindrical symmetry with respect to the axis whenever <math>A_\rho, A_\phi,</math> and <math> A_z</math> have this symmetry (no dependence on <math>\phi</math>); and it has reflectional symmetry only when <math>A_\phi = 0</math>.

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 [[Euclidean group#Subgroups|subgroups of the Euclidean group]].

==Symmetry groups in general==
{{see also|Automorphism|Automorphism group}}
In wider contexts, a '''symmetry group''' may be any kind of '''transformation group''', or [[automorphism]] group. Each type of [[mathematical structure]] has [[Bijection|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 group|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 [[M. C. Escher|Escher]].) Similarly, automorphism groups of [[finite geometry|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 [[Graph (discrete mathematics)|combinatorial graph]]: a graph symmetry is a permutation of the vertices which takes edges to edges. Any [[Presentation of a group|finitely presented group]] is the symmetry group of its [[Cayley graph]]; the [[free group]] is the symmetry group of an infinite [[Tree (graph theory)|tree graph]].

==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]], and let ''H'' &sub; ''G'' be a subgroup. Then ''H'' can be interpreted as the symmetry group of ''X''<sup>+</sup>, 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''<sup>#</sup> with Sym(''X''<sup>#</sup>) = {1}, the trivial subgroup; that is, ''gX''<sup>#</sup> &ne; ''X''<sup>#</sup> for all non-trivial ''g'' &isin; ''G''. Now we get:
:<math>
X^+ \ = \ \bigcup_{h\in H} hX^{\#} \quad\text{satisfies}\quad 
H = \mathrm{Sym}(X^+).
</math>

[[Normal subgroup]]s may also be characterized in this framework. 
The symmetry group of the translation ''gX'' <sup>+</sup> is the conjugate subgroup ''gHg''<sup>&minus;1</sup>. Thus ''H'' is normal whenever:
:<math> 
\mathrm{Sym}(gX^+) = \mathrm{Sym}(X^+) \ \ \text{for all} \ g\in G;
</math>
that is, whenever the decoration of ''X''<sup>+</sup> may be drawn in any orientation, with respect to any side or feature of ''X'', and still yield the same symmetry group ''gHg''<sup>&minus;1</sup> = ''H''.

As an example, consider the dihedral group ''G'' = ''D''<sub>3</sub> = Sym(''X''), where ''X'' is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure ''X''<sup>#</sup>. Letting &tau; &isin; ''G'' be the reflection of the arrowed edge, the composite figure ''X''<sup>+</sup> = ''X''<sup>#</sup> &cup; &tau;''X''<sup>#</sup> has a bidirectional arrow on that edge, and its symmetry group is ''H'' = {1, &tau;}. This subgroup is not normal, since ''gX''<sup>+</sup> may have the bi-arrow on a different edge, giving a different reflection symmetry group.

However, letting H = {1, &rho;, &rho;<sup>2</sup>} &sub; ''D''<sub>3</sub> be the cyclic subgroup generated by a rotation, the decorated figure ''X''<sup>+</sup> 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''.

==See also==
{{columns-list|colwidth=22em|
*[[Crystal system]]
*[[Euclidean plane isometry]]
*[[Fixed points of isometry groups in Euclidean space]]
*[[Molecular symmetry]]
*[[Permutation group]]
*[[Symmetric group]]
*[[Symmetry in quantum mechanics]]
}}

==Further reading==
*{{cite book | last1 = Burns | first1 = G. | last2 = Glazer | first2 = A. M. | year = 1990 | title = Space Groups for Scientists and Engineers | edition = 2nd | location = Boston | publisher = Academic Press, Inc | isbn = 0-12-145761-3 | url-access = registration | url = https://archive.org/details/spacegroupsforso0000burn }}
* {{cite book | last = Clegg | first = W | year = 1998 | title = Crystal Structure Determination (Oxford Chemistry Primer) | publisher = [[Oxford University Press]] | location = Oxford | isbn = 0-19-855901-1}}
*{{cite book  | last1=O'Keeffe | first1=M. | last2=Hyde |first2=B. G. | year = 1996 | title = Crystal Structures; I. Patterns and Symmetry | publisher = Mineralogical Society of America, Monograph Series | location = Washington, DC | isbn = 0-939950-40-5}}
*{{cite book |last=Miller |first=Willard Jr. |year=1972 |title=Symmetry Groups and Their Applications |publisher=Academic Press |location=New York |oclc=589081 |url=http://www.ima.umn.edu/~miller/symmetrygroups.html |access-date=2009-09-28 |url-status=dead |archive-url=https://web.archive.org/web/20100217091244/http://www.ima.umn.edu/~miller/symmetrygroups.html |archive-date=2010-02-17 }}

==External links==
* {{MathWorld | urlname=SymmetryGroup | title=Symmetry Group}}
* {{MathWorld | urlname=TetrahedralGroup | title=Tetrahedral Group}}
*[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html 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

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[[Category:Geometry]]
[[Category:Symmetry]]
[[Category:Group theory]]