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In mathematics, the isometry group of a metric space is the set of all isometries 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

• Consider a triangle in the plane with unequal sides. Then, the isometry group of the set of three vertices of this triangle is the trivial group. If the triangle has two equal sides which are not equal to the third, the isometry group is the cyclic group Z/2Z. If the triangle is equilateral, its isometry group is the permutation group S₃.

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

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

See also

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