Isometry group

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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.

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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

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Discrete group

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Symmetry (geometry)

In geometry, an object has symmetry if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

References

  1. 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 .
  2. 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 .
  3. 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 .
  4. 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 .