In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric...

composition, called the isometry group. There is also the weaker notion of path isometry or arcwise isometry: A path isometry or arcwise isometry is a map which...

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations...

the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient...

the Bohr compactification, and in group cohomology theory of Lie groups. A discrete isometry group is an isometry group such that for every point of the...

36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18°...

A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set...

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...

In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale...

The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It...

as abstract groups. Dih(S1), or orthogonal group O(2,R), or O(2): the isometry group of a circle, or equivalently, the group of isometries in 2D that keep...

§ Classification). The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated by reflections...

In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable...

Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that...

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

distance-preserving surjective map (that is, an isometry of metric spaces) between two connected Riemannian manifolds is a smooth isometry of Riemannian manifolds. A simpler...

the distance between each pair of points (an isometry). The corresponding group is called isometry group of X. If instead angles are preserved, one speaks...

isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry: genus 0 – the isometry group of...

d: the group G = A4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral...

theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically...

to their symmetry group type. Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information)...

automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one...

whose isometry group is the complex automorphism group SL(2, R) = Sp(2, R), the Siegel upper half-space has only one metric up to scaling whose isometry group...

space form, modulo a certain group action of isometries. For example, the isometry group of the n-sphere is the orthogonal group O(n + 1). Given any finite...

when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism...

Its isometry group is a compact orthogonal group O(n). This stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite...

fundamental group in the isometry group of the hyperbolic plane, which is isomorphic to PSL2(R) and this realizes the fundamental group as a Fuchsian group. A...

a group Centralizer and normalizer Characteristic subgroup Commutator Composition series Conjugacy class Conjugate closure Conjugation of isometries in...

Isometry group Quasi-isometry Dade isometry Euclidean isometry Euclidean plane isometry Itō isometry Isometric (disambiguation) Isometries in physics...

in terms of isometries of Euclidean space. A set of isometries can form a group. By a discrete isometry group we will mean an isometry group that maps each...