- Isometry group
In

mathematics , the**isometry group**of ametric space is the set of all isometries from the metric space onto itself, with thefunction composition as group operation. Itsidentity element is theidentity 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. Seesymmetry 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 thetrivial group . If the triangle has two equal sides which are not equal to the third, the isometry group is thecyclic group **Z**/2**Z**. If the triangle is equilateral, its isometry group is thepermutation group "S"_{3}.* The isometry group of a two-dimensional

sphere is an infinite group, called theorthogonal group "O"(3).* The isometry group of the "n"-dimensional

Euclidean space is theEuclidean group "E"(n).**ee also***

point groups in two dimensions

*point groups in three dimensions

*fixed points of isometry groups in Euclidean space

*Wikimedia Foundation.
2010.*