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Symmetry group

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Not to be confused withSymmetric group.

This article is about symmetry groups of geometric objects. For other uses, seeSymmetry group (disambiguation).

Group of transformations under which the object is invariant

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A regulartetrahedron is invariant under twelve distinctrotations (if the identity transformation is included as a trivial rotation and reflections are excluded). These are illustrated here in thecycle graph format, along with the 180° edge (blue arrows) and 120° vertex (pink and orange arrows) rotations thatpermute the tetrahedron through the positions. The twelve rotations form therotation (symmetry) group of the figure.

Ingroup theory, thesymmetry group of a geometric object is thegroup of alltransformations under which the object isinvariant, endowed with the group operation ofcomposition. Such a transformation is an invertible mapping of theambient 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 objectX isG = Sym(X).

For an object in ametric space, its symmetries form asubgroup of theisometry group of the ambient space. This article mainly considerssymmetry groups inEuclidean geometry, but the concept may also be studied for more general types of geometric structure.

Introduction

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We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as awallpaper 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, a function of position with values in a set of colors or substances; as avector field; or as a more general function on the object.) The group of isometries of space induces agroup action on objects in it, and the symmetry group Sym(X) consists of those isometries which mapX to itself (as well as mapping any further pattern to itself). We sayX isinvariant under such a mapping, and the mapping is asymmetry ofX.

The above is sometimes called thefull symmetry group ofX to emphasize that it includes orientation-reversing isometries (reflections,glide reflections andimproper rotations), as long as those isometries map this particularX to itself. The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called itsproper symmetry group. An object ischiral when it has noorientation-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.

Any symmetry group whose elements have a commonfixed point, which is true if the group is finite or the figure is bounded, can be represented as a subgroup of theorthogonal 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 therotation group of the figure.

In adiscrete symmetry group, the points symmetric to a given point do not accumulate toward alimit point. That is, everyorbit of the group (the images of a given point under all group elements) forms adiscrete set. All finite symmetry groups are discrete.

Discrete symmetry groups come in three types: (1) finitepoint groups, which include onlyrotations,reflections, inversions androtoinversions – i.e., the finite subgroups of O(n); (2) infinitelattice groups, which include only translations; and (3) infinitespace groups containing elements of both previous types, and perhaps also extra transformations likescrew displacements and glide reflections. There are alsocontinuous symmetry groups (Lie groups), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example isO(3), the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as thesubgroups of the Euclidean group E(n) (the isometry group ofRⁿ).

Two geometric figures have the samesymmetry type when their symmetry groups areconjugate subgroups of the Euclidean group: that is, when the subgroupsH₁,H₂ are related byH₁ =g⁻¹H₂g for someg in E(n). For example:

two 3D figures have mirror symmetry, but with respect to different mirror planes.
two 3D figures have 3-foldrotational symmetry, but with respect to different axes.
two 2D patterns havetranslational 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 whoseorbits aretopologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by arational number; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.

One dimension

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Main article:One-dimensional symmetry group

The isometry groups in one dimension are:

the trivialcyclic group C₁
the groups of two elements generated by a reflection; they areisomorphic with C₂
the infinite discrete groups generated by a translation; they are isomorphic withZ, the additive group of the integers
the infinite discrete groups generated by a translation and a reflection; they are isomorphic with thegeneralized dihedral group ofZ, Dih(Z), also denoted by D_∞ (which is asemidirect product ofZ and C₂).
the group generated by all translations (isomorphic with the additive group of the real numbersR); 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

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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 groups D₁, D₂,D₃,D₄, ..., where D_n (of order 2n) consists of the rotations in C_n together with reflections inn axes that pass through the fixed point.

C₁ is thetrivial group 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 atriskelion, C₄ of aswastika, 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 ofbilateral symmetry, for example the letter "A".

D₂, which is isomorphic to theKlein 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.

D₃, D₄ etc. are the symmetry groups of theregular polygons.

Within each of these symmetry types, there are twodegrees 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 thecircle group S¹, the multiplicative group ofcomplex numbers ofabsolute value 1. It is theproper symmetry group of a circle and the continuous equivalent of C_n. There is no geometric figure that has asfull 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 7frieze groups
the 17wallpaper 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.

Three dimensions

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Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. Incrystallography, only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). Thiscrystallographic restriction of the infinite families of general point groups results in 32crystallographic 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 abottle orcone.
cylindrical symmetry with a symmetry plane perpendicular to the axis
spherical symmetry

For objects withscalar field patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true forvector field patterns: for example, incylindrical coordinates with respect to some axis, the vector field $\mathbf {A} =A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$ 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 ascrew axis, such as an infinitehelix. See alsosubgroups of the Euclidean group.

Symmetry groups in general

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See also:Automorphism andAutomorphism group

In wider contexts, asymmetry group may be any kind oftransformation group, orautomorphism group. Each type ofmathematical structure hasinvertible 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 theErlangen programme.

For example, objects in a hyperbolicnon-Euclidean geometry haveFuchsian 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 ofEscher.) Similarly, automorphism groups offinite 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 acombinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Anyfinitely presented group is the symmetry group of itsCayley graph; thefree group is the symmetry group of an infinitetree graph.

Group structure in terms of symmetries

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Cayley's theorem states that any abstract group is a subgroup of the permutations of some setX, and so can be considered as the symmetry group ofX 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, letG = Sym(X) be the finite symmetry group of a figureX in aEuclidean space, and letH ⊂G be a subgroup. ThenH can be interpreted as the symmetry group ofX⁺, a "decorated" version ofX. Such a decoration may be constructed as follows. Add some patterns such as arrows or colors toX so as to break all symmetry, obtaining a figureX^# with Sym(X^#) = {1}, the trivial subgroup; that is,gX^# ≠X^# for all non-trivialg ∈G. Now we get:

X^{+}\ =\ \bigcup _{h\in H}hX^{\#}\quad {\text{satisfies}}\quad H=\mathrm {Sym} (X^{+}).

Normal subgroups may also be characterized in this framework. The symmetry group of the translationgX⁺ is the conjugate subgroupgHg⁻¹. ThusH is normal whenever:

\mathrm {Sym} (gX^{+})=\mathrm {Sym} (X^{+})\ \ {\text{for all}}\ g\in G;

that is, whenever the decoration ofX⁺ may be drawn in any orientation, with respect to any side or feature ofX, and still yield the same symmetry groupgHg⁻¹ =H.

As an example, consider the dihedral groupG =D₃ = Sym(X), whereX is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figureX^#. Letting τ ∈G be the reflection of the arrowed edge, the composite figureX⁺ =X^# ∪ τX^# has a bidirectional arrow on that edge, and its symmetry group isH = {1, τ}. This subgroup is not normal, sincegX⁺ may have the bi-arrow on a different edge, giving a different reflection symmetry group.

However, letting H = {1, ρ, ρ²} ⊂D₃ be the cyclic subgroup generated by a rotation, the decorated figureX⁺ consists of a 3-cycle of arrows with consistent orientation. ThenH is normal, since drawing such a cycle with either orientation yields the same symmetry groupH.