Isometry Group - Examples

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

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