Isometry group

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 space onto itself, with the function composition as group operation.¹ Its identity element is the identity function.² The elements of the isometry group are sometimes called motions of the space.

Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See symmetry group.

A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set.

In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to form an isometry group of the pseudo-Euclidean space.

• The isometry group of the subspace of a metric space consisting of the points of a scalene triangle is the trivial group. A similar space for an isosceles triangle is the cyclic group of order two, C₂. A similar space for an equilateral triangle is D₃, the dihedral group of order 6.
• The isometry group of a two-dimensional sphere is the orthogonal group O(3).³

• The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n).⁴
• The isometry group of the Poincaré disc model of the hyperbolic plane is the projective special unitary group PSU(1,1).
• The isometry group of the Poincaré half-plane model of the hyperbolic plane is PSL(2,R).
• The isometry group of Minkowski space is the Poincaré group.⁵

• Riemannian symmetric spaces are important cases where the isometry group is a Lie group.