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The symmetry group of a geometric figure is the group of congruencies 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 orientation-reversing congruencies (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving 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.
The discrete point groups in 2 dimensional space consist of two infinite families
- cyclic groups C1, C2, C3, ... where Cn consists of all rotations about a fixed point by multiples of the angle 2π/n
- dihedral groups D1, D2, D3, ... where Dn consists of the rotations in Cn 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 H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1=g-1H2g. Up to conjugacy the finite subgroups of O(2) are the cyclic groups and the dihedral groups.
C1 is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C2 is the symmetry group of the letter Z, C3 that of a triskelion, C4 of a swastika, 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, for example the letter A. D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D3, D4 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 non-drawable 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.
The situation in 3-D 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 Cs (or C1h), Ci, and C2. 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 Cn, 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 Cnh, or a set of n mirror planes containing the axis, giving the group Cnv.
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 Dnh. Its subgroup of rotations called Dn still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the main rotation axis but located halfway between the other 2-fold axes, so the perpendicular plane is not there. Dnh and Dnd are the symmetry groups for regular prisms and antiprisms, respectively. Dn is the symmetry group of a partially rotated prism.
There is one more group in this family to mention, called Sn. 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 Cnh, 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
- Cs, C1h and C1v are all groups of order 2 with a single reflection
- C2 and D1 are groups of order 2 with a single 180° rotation
- Ci and S2 are groups of order 2 with a single inversion
- D1h and C2v are groups of order 4 with a reflection in a plane π and a 180° rotation through a line in π
- D1d and C2h 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 Cn to denote an axis of rotation through 2π/n and Sn to denote an axis of improper rotation through the same, the groups are:
- T (tetrahedral). There are four C3 axes, directed through the corners of a cube, and three C2 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 A4, the alternating group on 4 letters, and is the rotation group for a regular tetrahedron.
- Td. This group has the same rotation axes as T, but with six mirror planes, each containing a single C2 axis and four C3 axes. The C2 axes are now actually S4 axes. This group has order 24, and is the symmetry group for a regular tetrahedron. Td is isomorphic to S4, the symmetric group on 4 letters.
- Th. This group has the same rotation axes as T, but with mirror planes, each containing two C2 axes and no C3 axes. The C3 axes become S6 axes, and a center of inversion appears. Again, group has order 24. Th is isomorphic to A4 × C2.
- O (octahedral). This group is similar to T, but the C2 axes are now C4 axes, and a new set of 12 C2 axes appear, directed towards the edges of the original cube. This group of order 24 is also isomorphic to S4, and is the rotation group of the cube and octahedron.
- Oh. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group has order 48, is isomorphic to S4 × C2, and is the symmetry group of the cube and octahedron.
- I, Ih (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 Ih having order 120. The group I is isomorphic to A5, the alternating group on 5 letters, and Ih to A5 × C2....
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 C1, C2, C3 etc., the dihedral groups D2, D3, D4 etc., and the rotation groups T, O and I of a regular tetrahedron, octahedron and icosahedron.
In particular, the dihedral groups D3, D4 etc. are the rotation groups of plane regular polygons embedded in three-dimensional 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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