Symmetry group

A symmetry group is the group of transformations that preserve a specified object or structure. Its operations capture the object's symmetries and obey the group axioms.

Group action: A group action assigns each group element a transformation of a set, compatibly with the group operation. Symmetries act on the object's points, states, or other structural elements.
Group representation: A group representation realizes group elements as invertible linear transformations of a vector space. Representations turn abstract symmetries into matrices that can be calculated and compared.
Noether's theorem: Noether's theorem links continuous symmetries of an action to conserved quantities in physical systems. Continuous symmetry groups explain conservation laws such as energy and momentum.
Symmetry (physics): Symmetry in physics is invariance of a system's laws or state under a specified transformation. The symmetry group is the algebraic collection of transformations behind this broader physical idea.
Group: A group is a set with an associative operation, an identity element, and inverses for every element. A symmetry group is a group whose elements are structure-preserving transformations.
Permutation group: A permutation group is a group of bijections of a set onto itself. Symmetries of finite objects often act by permuting their vertices, edges, or components.
Crystallographic point group: A crystallographic point group is a point group compatible with the periodicity of a crystal lattice. These groups classify the possible local symmetries of crystalline materials.
Approximate symmetry: An approximate symmetry is a transformation that preserves a system's structure only within a limited accuracy or regime. Real systems may nearly, but not exactly, satisfy the preservation condition defining a symmetry group.
Invariant (mathematics): An invariant is a property or quantity unchanged by specified transformations. The defining condition for each symmetry is preservation of the structure's relevant features.
Dihedral group: The dihedral group is the group of rotations and reflections preserving a regular polygon. It is a standard example where rotations and reflections combine into a symmetry group.

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