Symmetry group

(This page currently does not yet describe various aspects of symmetry groups in theoretical physics, especially in (quantum and classical) field theory.)
The symmetry group of a geometric shape is intuitively: all transformations that can be applied to a figure without changing it (i.e. the transformation rotate is part of the symmetry group for a circle). More formally, the symmetry group of a geometric figure is the group of congruencies (i.e. transformations) under which it is invariant, with composition as the operation. (If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts.)
The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientationreversing congruencies (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientationpreserving congruencies (i.e. translations, rotations and compositions of these) which leave the figure invariant is called proper symmetry group.
Any symmetry group whose elements have a common fixed point can be represented as a subgroup of O(n) (by choosing the origin to be a fixed point). This is true for all finite symmetry groups, and also for the symmetry groups of bounded figures. Of course, the proper symmetry group is a subgroup of SO(n) then, and therefore also called rotation group of the figure.
Discrete symmetry groups come in two types: finite point groups, which include only rotations and reflections  they are in fact just the finite subgroups of O(n), and infinite lattice groups, which also include translations and possibly glide reflections. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups.
Contents 
Two dimensions
The discrete point groups in 2 dimensional space consist of two infinite families
 cyclic groups C_{1}, C_{2}, C_{3}, ... where C_{n} consists of all rotations about a fixed point by multiples of the angle 2π/n
 dihedral groups D_{1}, D_{2}, D_{3}, ... where D_{n} consists of the rotations in C_{n} together with reflections in n axes that pass through the fixed point.
More precisely, this is a classification up to conjugacy. Remember that two subgroups H_{1}, H_{2} of a group G are conjugate, if there exists g ∈ G such that H_{1}=g^{1}H_{2}g. Up to conjugacy the finite subgroups of O(2) are the cyclic groups and the dihedral groups.
C_{1} is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, 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 swastikalike figures with five, six etc. arms instead of four.
D_{1} is the 2element 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 fourgroup, is the symmetry group of a nonequilateral rectangle, and D_{3}, D_{4} etc. are the symmetry groups of the regular polygons.
Providing the figure is bounded and topologically closed (so that the group is a complete point group) the only other possibility is the 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. The closure condition here is a natural one for subsets of the plane that can be considered "figures", as it excludes nondrawable sets such as the set of all points on the unit circle with rational coordinates. The symmetry group of this set includes some, but not all, arbitrarily small rotations.
For nonbounded figures, the symmetry group can include translations, so that the seventeen wallpaper groups and seven frieze groups are possibilities.
Three dimensions
The situation in 3D is more complicated, since it is possible to have multiple rotation axes in a point group. First, of course, there is the trivial group, and then there are three groups of order 2, called C_{s} (or C_{1h}), C_{i}, and C_{2}. These have the single symmetry operation of reflection in a plane, in a point, and in a line (equivalent to a rotation of π), respectively.
The last of these is the first of the uniaxial groups C_{n}, which are generated by a single rotation of angle 2π/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group C_{nh}, or a set of n mirror planes containing the axis, giving the group C_{nv}.
If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through π, so the group is no longer uniaxial. This new group is called D_{nh}. Its subgroup of rotations called D_{n} still has the 2fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. There is one more group in this family, called D_{nd} (or D_{nv}), which has vertical mirror planes containing the main rotation axis but located halfway between the other 2fold axes, so the perpendicular plane is not there. D_{nh} and D_{nd} are the symmetry groups for regular prisms and antiprisms, respectively. D_{n} is the symmetry group of a partially rotated prism.
There is one more group in this family to mention, called S_{n}. This group is generated by an improper rotation of angle 2π/n  that is, a rotation followed by a reflection about a plane perpendicular to its axis. For n odd, the rotation and reflection are generated, so this becomes the same as C_{nh}, but it remains distinct for n even.
Of these 7 families of groups described above, it has been noted that there are some coincidences for small values of n. In particular
 C_{s}, C_{1h} and C_{1v} are all groups of order 2 with a single reflection
 C_{2} and D_{1} are groups of order 2 with a single 180° rotation
 C_{i} and S_{2} are groups of order 2 with a single inversion
 D_{1h} and C_{2v} are groups of order 4 with a reflection in a plane π and a 180° rotation through a line in π
 D_{1d} and C_{2h} are groups of order 4 with a reflection in a plane π and a 180° rotation through a line perpendicular to π
The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using C_{n} to denote an axis of rotation through 2π/n and S_{n} to denote an axis of improper rotation through the same, the groups are:
 T (tetrahedral). There are four C_{3} axes, directed through the corners of a cube, and three C_{2} axes, directed through the centers of its faces. There are no other symmetry operations, giving the group an order of 12. This group is isomorphic to A_{4}, the alternating group on 4 letters, and is the rotation group for a regular tetrahedron.
 T_{d}. This group has the same rotation axes as T, but with six mirror planes, each containing a single C_{2} axis and four C_{3} axes. The C_{2} axes are now actually S_{4} axes. This group has order 24, and is the symmetry group for a regular tetrahedron. T_{d} is isomorphic to S_{4}, the symmetric group on 4 letters.
 T_{h}. This group has the same rotation axes as T, but with mirror planes, each containing two C_{2} axes and no C_{3} axes. The C_{3} axes become S_{6} axes, and a center of inversion appears. Again, group has order 24. T_{h} is isomorphic to A_{4} × C_{2}.
 O (octahedral). This group is similar to T, but the C_{2} axes are now C_{4} axes, and a new set of 12 C_{2} axes appear, directed towards the edges of the original cube. This group of order 24 is also isomorphic to S_{4}, and is the rotation group of the cube and octahedron.
 O_{h}. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_{d} and T_{h}. This group has order 48, is isomorphic to S_{4} × C_{2}, and is the symmetry group of the cube and octahedron.
 I, I_{h} (icosahedral) are the groups of symmetries of the icosahedron and the dodecahedron. The group of proper rotations, I, is a normal subgroup of index 2 in the full group of symmetries, with I having order 60 and I_{h} having order 120. The group I is isomorphic to A_{5}, the alternating group on 5 letters, and I_{h} to A_{5} × C_{2}....
This completes the classification (up to conjugacy) of the finite subgroups of O(3).
The classification of the finite subgroups of SO(3), which occur as rotation groups of bounded figures, is easier: Up to conjugacy the finite subgroups of SO(3) are the follwing classes: the cyclic groups C_{1}, C_{2}, C_{3} etc., the dihedral groups D_{2}, D_{3}, D_{4} etc., and the rotation groups T, O and I of a regular tetrahedron, octahedron and icosahedron.
In particular, the dihedral groups D_{3}, D_{4} etc. are the rotation groups of plane regular polygons embedded in threedimensional space, and such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.
Symmetry groups in general
In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme.
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