# 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.[1] 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.

## 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 two, C
_{2}. A similar space for an equilateral triangle is D_{3}, the dihedral group of order 6. - The isometry group of a two-dimensional sphere is the orthogonal group O(3).[2]

- The isometry group of the
*n*-dimensional Euclidean space is the Euclidean group E(*n*).[3] - The isometry group of the Poincaré disc model of the hyperbolic plane is SU(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.[4]

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

## See also

## References

- Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001),
*A course in metric geometry*, Graduate Studies in Mathematics,**33**, Providence, RI: American Mathematical Society, p. 75, ISBN 0-8218-2129-6, MR 1835418. - Berger, Marcel (1987),
*Geometry. II*, Universitext, Berlin: Springer-Verlag, p. 281, doi:10.1007/978-3-540-93816-3, ISBN 3-540-17015-4, MR 0882916. - Olver, Peter J. (1999),
*Classical invariant theory*, London Mathematical Society Student Texts,**44**, Cambridge: Cambridge University Press, p. 53, doi:10.1017/CBO9780511623660, ISBN 0-521-55821-2, MR 1694364. - Müller-Kirsten, Harald J. W.; Wiedemann, Armin (2010),
*Introduction to supersymmetry*, World Scientific Lecture Notes in Physics,**80**(2nd ed.), Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., p. 22, doi:10.1142/7594, ISBN 978-981-4293-42-6, MR 2681020.