Isometry group, the Glossary

Table of Contents

1. 42 relations: Bijection, Cyclic group, Dihedral group of order 6, Equilateral triangle, Euclidean group, Euclidean space, Fixed points of isometry groups in Euclidean space, Function composition, Graduate Studies in Mathematics, Graduate Texts in Mathematics, Group (mathematics), Hyperbolic geometry, Identity element, Identity function, Isolated point, Isometry, Isosceles triangle, Isotropic quadratic form, Lie group, Linear subspace, Mathematics, Metric space, Minkowski space, Motion (geometry), Order (group theory), Orthogonal group, Poincaré disk model, Poincaré group, Poincaré half-plane model, Point group, Point groups in three dimensions, Point groups in two dimensions, Projective unitary group, Pseudo-Euclidean space, Set (mathematics), SL2(R), Sphere, Subgroup, Symmetric space, Symmetry, Symmetry group, Trivial group.

Bijection

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain).

See Isometry group and Bijection

Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

See Isometry group and Cyclic group

Dihedral group of order 6

In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6.

See Isometry group and Dihedral group of order 6

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have the same length.

See Isometry group and Equilateral triangle

Euclidean group

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations).

See Isometry group and Euclidean group

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

See Isometry group and Euclidean space

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.

See Isometry group and Fixed points of isometry groups in Euclidean space

Function composition

In mathematics, function composition is an operation that takes two functions and, and produces a function such that.

See Isometry group and Function composition

Graduate Studies in Mathematics

Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).

See Isometry group and Graduate Studies in Mathematics

Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

See Isometry group and Graduate Texts in Mathematics

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Isometry group and Group (mathematics)

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. Isometry group and hyperbolic geometry are metric geometry.

See Isometry group and Hyperbolic geometry

Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied.

See Isometry group and Identity element

Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged.

See Isometry group and Identity function

Isolated point

In mathematics, a point is called an isolated point of a subset (in a topological space) if is an element of and there exists a neighborhood of that does not contain any other points of.

See Isometry group and Isolated point

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. Isometry group and isometry are metric geometry.

See Isometry group and Isometry

Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

See Isometry group and Isosceles triangle

Isotropic quadratic form

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero.

See Isometry group and Isotropic quadratic form

Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

See Isometry group and Lie group

Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspaceThe term linear subspace is sometimes used for referring to flats and affine subspaces.

See Isometry group and Linear subspace

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Isometry group and Mathematics

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Isometry group and Metric space

Minkowski space

In physics, Minkowski space (or Minkowski spacetime) is the main mathematical description of spacetime in the absence of gravitation.

See Isometry group and Minkowski space

Motion (geometry)

In geometry, a motion is an isometry of a metric space. Isometry group and motion (geometry) are metric geometry.

See Isometry group and Motion (geometry)

Order (group theory)

In mathematics, the order of a finite group is the number of its elements.

See Isometry group and Order (group theory)

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.

See Isometry group and Orthogonal group

Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.

See Isometry group and Poincaré disk model

Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime.

See Isometry group and Poincaré group

Poincaré half-plane model

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H.

See Isometry group and Poincaré half-plane model

Point group

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common.

See Isometry group and Point group

Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.

See Isometry group and Point groups in three dimensions

Point groups in two dimensions

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane.

See Isometry group and Point groups in two dimensions

Projective unitary group

In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center,, embedded as scalars.

See Isometry group and Projective unitary group

Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space of signature is a finite-dimensional ''n''-space together with a non-degenerate quadratic form.

See Isometry group and Pseudo-Euclidean space

Set (mathematics)

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

See Isometry group and Set (mathematics)

SL2(R)

In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \colon a,b,c,d \in \mathbf\mboxad-bc.

See Isometry group and SL2(R)

Sphere

A sphere (from Greek) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.

See Isometry group and Sphere

Subgroup

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

See Isometry group and Subgroup

Symmetric space

In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point.

See Isometry group and Symmetric space

Symmetry

Symmetry in everyday life refers to a sense of harmonious and beautiful proportion and balance.

See Isometry group and Symmetry

Symmetry group

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition.

See Isometry group and Symmetry group

Trivial group

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

See Isometry group and Trivial group

References

[1] https://en.wikipedia.org/wiki/Isometry_group

Also known as Discrete isometry group, Isometry Groups.