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# Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation. 

72 relations: Absolute value, Automorphism, Bipyramid, Bottle, Chirality (mathematics), Circle group, Closed set, Complex number, Continuous symmetry, Crystal system, Crystallographic point group, Crystallographic restriction theorem, Crystallography, Cube, Cyclic group, Cylindrical coordinate system, Degrees of freedom (physics and chemistry), Dihedral group, Dihedral group of order 6, Erlangen program, Euclidean geometry, Euclidean group, Euclidean plane isometry, Finite geometry, Fixed point (mathematics), Fixed points of isometry groups in Euclidean space, Frieze group, Function composition, Geometric transformation, Glide reflection, Group (mathematics), Group action, Group theory, Helix, If and only if, Image, Improper rotation, Invariant (mathematics), Isolated point, Isometry group, Klein four-group, Lagrange's theorem (group theory), Lattice (group), Lie group, Map (mathematics), Mathematical structure, Metric (mathematics), Molecular symmetry, One-dimensional symmetry group, Orientation (vector space), ... Expand index (22 more) »

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.

## Bottle

A bottle is a narrow-necked container as compared with a jar.

## Chirality (mathematics)

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone.

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C&times;, the multiplicative group of all nonzero complex numbers.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Continuous symmetry

In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another.

## Crystal system

In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals.

## Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind.

## Crystallographic restriction theorem

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold.

## Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure).

## Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

## Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

## Cylindrical coordinate system

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.

## Degrees of freedom (physics and chemistry)

In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system.

## Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

## Dihedral group of order 6

In mathematics, the smallest non-abelian group has 6 elements.

## Erlangen program

The Erlangen program is a method of characterizing geometries based on group theory and projective geometry.

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Euclidean group

In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.

## Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.

## Finite geometry

A finite geometry is any geometric system that has only a finite number of points.

## Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

## Fixed points of isometry groups in Euclidean space

A fixed point of an isometry group is a point that is a fixed point for every isometry in the group.

## Frieze group

In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Geometric transformation

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set.

## Glide reflection

In 2-dimensional geometry, a glide reflection (or transflection) is a type of opposite isometry of the Euclidean plane: the composition of a reflection in a line and a translation along that line.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Helix

A helix, plural helixes or helices, is a type of smooth space curve, i.e. a curve in three-dimensional space.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Image

An image (from imago) is an artifact that depicts visual perception, for example, a photo or a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person, thus providing a depiction of it.

## Improper rotation

In geometry, an improper rotation,.

## Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

## Isolated point

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).

## Isometry group

In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.

## Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

## Mathematical structure

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

## Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

## Molecular symmetry

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry.

## One-dimensional symmetry group

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D).

## Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

## Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).

## Point group

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.

## Reflection symmetry

Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.

## Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

## Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

## Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.

## Screw axis

A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs.

## Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.

## Signal

A signal as referred to in communication systems, signal processing, and electrical engineering is a function that "conveys information about the behavior or attributes of some phenomenon".

## Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Swastika

The swastika (as a character 卐 or 卍) is a geometrical figure and an ancient religious icon from the cultures of Eurasia, where it has been and remains a symbol of divinity and spirituality in Indian religions, Chinese religions, Mongolian and Siberian shamanisms.

## Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

## Symmetry

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.

## Symmetry in quantum mechanics

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics.

## Triskelion

A triskelion or triskele is a motif consisting of a triple spiral exhibiting rotational symmetry.

## Trivial group

In mathematics, a trivial group is a group consisting of a single element.

## Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Wallpaper group

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern.

## References

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