Isometry group
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In mathematics, the isometry group of a metric space is the set of all isometries (i.e. distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function.
A single 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.
Examples
- The isometry group of the space consisting of three points, with a metric as for the vertices of a triangle in the plane with all sides unequal, is the trivial group. With a metric as for a triangle with two equal sides that are unequal to the third, it is the cyclic group of order 2, C2, and 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
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