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.