Definition. A space group in n dimensions is a subgroup of En. Definition. The point group of a given space group is the subgroup of symmetry operations that leave one point fixed (i.e. proper and improper rotations). In other words, the point group of a space group is its intersection with O(n).Jun 8, 2011.
What is group and point group?
A Point Group describes all the symmetry operations that can be performed on a molecule that result in a conformation indistinguishable from the original. Point groups are used in Group Theory, the mathematical analysis of groups, to determine properties such as a molecule's molecular orbitals..
What is the difference between space groups and point groups?
A space group in n dimensions is a subgroup of En. The point group of a given space group is the subgroup of symmetry operations that leave one point fixed (i.e. proper and improper rotations). In other words, the point group of a space group is its intersection with O(n).Jun 8, 2011.
A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations.
Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).
The crystallographic point group is a set of symmetry operations that leave at least one point unmoved.A space group is the 3D symmetry group of a configuration in space. There are 32 crystallographic point groups. There are 230 space groups (created by the combination of 32 point groups and 14 Bravais lattices).
Overview
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions
Notation
The point groups are named according to their component symmetries. There are several standard notations used by crystallographers
Isomorphisms
Many of the crystallographic point groups share the same internal structure. For example, the point groups 1, 2
Deriving the crystallographic point group (crystal class) from the space …
1. Leave out the Bravais lattice type
See also
• Molecular symmetry• Point group• Space
Crystallographic point groups and space groups
In mathematics, a topological group G is called a discrete group if there is no limit point in it. Equivalently, the group G is discrete if and only if its identity is isolated.
In mathematics
Type of symmetry group
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group.
Discrete subgroup of the real projective special linear group of dimension 2
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R), and sometimes it is allowed to be a Kleinian group which is conjugate to a subgroup of PSL(2,R).
