Editing Symmetry group (section)

==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.