Point group

A point group is a group of symmetry operations that leave at least one point fixed. Its operations describe symmetries that preserve a common center, such as rotations, reflections, and inversions.

Group theory: The mathematical study of sets equipped with associative operations, identity elements, and inverses. Point groups are groups whose elements act as geometric symmetries.
Rotation: A rigid transformation that turns points around an axis by a specified angle. Rotations generate many point groups and define their rotational axes.
Molecular symmetry: The set of spatial transformations that leave a molecule indistinguishable from its original configuration. A molecule's point group summarizes its full symmetry.
Space group: A group of symmetry operations describing a periodic structure, including translations. Unlike a point group, a space group can move every point through lattice translations.
Symmetry operation: A transformation that leaves an object or pattern indistinguishable from its original form. Each point-group element is a symmetry operation.
Reflection: A transformation that maps each point to its mirror image across a line or plane. Mirror planes supply symmetry operations that can combine with rotations.
Schoenflies notation: A notation system that labels molecular symmetry groups using symbols such as C₂v and D₃h. It names point groups commonly used in molecular chemistry.
Wallpaper group: A symmetry group describing repeating patterns in a plane, including translations. Its planar translations distinguish it from point groups.
Group action: A way for a group to act on a set through transformations that respect the group operation. It formalizes how point-group operations transform points in space.
Inversion symmetry: Symmetry under mapping every position vector to its negative through a fixed center. Inversion is a central operation found in many crystallographic point groups.

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