Symmetry group

In mathematical group theory, the symmetry group of a geometric object is the group that consists of the set of all Kongruenzabbildungen that map the object to itself, together with the concatenation of pictures as the group operation.

Disambiguation

The following terms describe possible properties of an object against which it can be determined which symmetry group the object belongs.

Discreetness

A symmetry group then has a discrete topology, if there is such a thing as "smallest steps ". For example, a group of turns about a point if and only discrete, if all possible angles of rotation are multiples of a smallest angle. If, however, even arbitrarily small rotation angle included in the group, this group is not discrete. In general, each group with finitely many elements a discrete topology. A discrete group can be generated from a finite number of symmetry operations by composition. The reverse does not apply in each case.

In practical terms a symmetry group is precisely then discretely when there is a lower bound for both the lengths of all the ( non-zero ) shifts and the rotation angles of all the rotational symmetry.

Periodicity

Considering the set of all included in the group ( other than zero ) displacements (translations ), and determines how many of these vectors are linearly independent from each other, thus determining the linear dimension of the sheath of the displacement vectors.

If the group contains no shifts, so there is at least one point is the fixed point of all the pictures. One speaks in this case of a point group. Point groups are exactly then finally, if they are discreet.

Once the group contains at least one shift, it automatically contains infinitely many elements at least in Euclidean geometry.

Corresponds to the number of linearly independent displacement vectors of the dimension of the space in which the object is embedded, it is a limited portion of the subject (a cell ), the images will fill the entire space. If the group is additionally also discreet, so we speak of a space group and is called the pattern periodically. In this case, there is a limited range of the fundamental of the same dimension as the space, so for example in the plane of a corresponding non-zero area.

Two-dimensional Euclidean geometry

The symmetry groups in the Euclidean plane can be classified as follows:

• Discrete Without shifts Without axis reflections Family of finite cyclic groups ( for ), which are all rotations about a point by multiples of: Symmetry group of a completely unbalanced object, with the identity as the only element: Symmetry group of a point reflection: Symmetry group of a Triskele: Symmetry group of a Swastika
• With axis reflections Family of dihedral ( for ), the rotations are as together with mirror axis through the center: Single axis mirroring: Symmetry group of a non-square rectangle: Symmetry group of a regular n -gon

• Without shifts Orthogonal group, which are all symmetries of a circle, so all rotations and reflections on all axes that pass through the center
• With shifts This case needs to be further broken down.

Other dimensions

• Three-dimensional point groups are classified in detail in a separate article.
• The article on space groups also addresses different dimensions.