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In mathematics, the isometry group of a metric space is the set of all isometries (i.e. injective, 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. ==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 2, C2. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「isometry group」の詳細全文を読む スポンサード リンク
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