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.
- 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 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 the hyperbolic plane is PGL(2,R) = PSL(2,R).2.
- point groups in two dimensions
- point groups in three dimensions
- fixed points of isometry groups in Euclidean space
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