Editing Isometry group (section)

{{Short description|Automorphism group of a metric space or pseudo-Euclidean space}}
In [[mathematics]], the '''isometry group''' of a [[metric space]] is the [[Set (mathematics)|set]] of all [[bijective]] [[isometry|isometries]] (that is, bijective, [[distance-preserving map]]s) from the metric space onto itself, with the [[function composition]] as [[group (mathematics)|group]] operation.<ref>{{citation |last=Roman |first=Steven |title=Advanced Linear Algebra |date=2008 |pages=271 |series=[[Graduate Texts in Mathematics]] |edition=Third |publisher=Springer |isbn=978-0-387-72828-5 |author-link=Steven Roman}}.</ref> Its [[identity element]] is the [[identity function]].<ref>{{citation
 | last1 = Burago | first1 = Dmitri
 | last2 = Burago | first2 = Yuri
 | last3 = Ivanov | first3 = Sergei
 | isbn = 0-8218-2129-6
 | mr = 1835418
 | page = 75
 | publisher = American Mathematical Society
 | location = Providence, RI
 | series = [[Graduate Studies in Mathematics]]
 | title = A course in metric geometry
 | url = https://books.google.com/books?id=afnlx8sHmQIC&pg=PA75
 | volume = 33
 | year = 2001}}.</ref> The elements of the isometry group are sometimes called [[motion (geometry)|motion]]s of the space.

Every isometry group of a metric space is a [[subgroup]] of isometries. It represents in most cases a possible set of [[symmetry|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.