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. Its identity element is the identity function.

A (generalized) isometry on a pseudo-Euclidean space preserves magnitude.

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.

Examples

• 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 2, C₂. As for an equilateral triangle, it is the dihedral group of order three, D₃.

• The isometry group of a two-dimensional sphere is the orthogonal group O(3).

• The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n).

• The isometry group of Minkowski space is the Poincaré group.

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

See also

• point groups in two dimensions
• point groups in three dimensions
• fixed points of isometry groups in Euclidean space