Editing Isometry group

{{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.

==Examples==

* The isometry group of the [[Linear subspace|subspace]] of a [[metric space]] consisting of the points of a [[Triangle#Types_of_triangle|scalene triangle]] is the [[trivial group]]. A similar space for an [[isosceles triangle]] is the [[cyclic group]] of [[Order (group theory)|order]] two, C<sub>2</sub>. A similar space for an [[equilateral triangle]] is D<sub>3</sub>, the [[dihedral group of order 6]]. 
* The isometry group of a two-dimensional [[sphere]] is the [[orthogonal group]] O(3).<ref>{{citation
 | last = Berger | first = Marcel
 | doi = 10.1007/978-3-540-93816-3
 | isbn = 3-540-17015-4
 | mr = 882916
 | page = 281
 | publisher = Springer-Verlag
 | location = Berlin
 | series = Universitext
 | title = Geometry. II
 | url = https://books.google.com/books?id=6WZHAAAAQBAJ&pg=PA281
 | year = 1987}}.</ref>

* The isometry group of the ''n''-dimensional [[Euclidean space]] is the [[Euclidean group]] E(''n'').<ref>{{citation
 | last = Olver | first = Peter J. |author-link=Peter J. Olver 
 | doi = 10.1017/CBO9780511623660
 | isbn = 0-521-55821-2
 | mr = 1694364
 | page = 53
 | publisher = Cambridge University Press
 | location = Cambridge
 | series = London Mathematical Society Student Texts
 | title = Classical invariant theory
 | url = https://books.google.com/books?id=1GlHYhNRAqEC&pg=PA53
 | volume = 44
 | year = 1999}}.</ref>
* The isometry group of the [[Poincaré disc model]] of the [[hyperbolic plane]] is the projective special unitary group [[Projective special unitary group|PSU(1,1)]].
* The isometry group of the [[Poincaré half-plane model]] of the hyperbolic plane is [[PSL(2,R)]].
* The isometry group of [[Minkowski space]] is the [[Poincaré group]].<ref>{{citation
 | last1 = Müller-Kirsten | first1 = Harald J. W.
 | last2 = Wiedemann | first2 = Armin
 | doi = 10.1142/7594
 | edition = 2nd
 | isbn = 978-981-4293-42-6
 | mr = 2681020
 | page = 22
 | publisher = World Scientific Publishing Co. Pte. Ltd.
 | location = Hackensack, NJ
 | series = World Scientific Lecture Notes in Physics
 | title = Introduction to supersymmetry
 | url = https://books.google.com/books?id=RU-hsrWp9isC&pg=PA22
 | volume = 80
 | year = 2010}}.</ref>

* [[Riemannian symmetric space]]s are important cases where the isometry group is a [[Lie group]].

==See also==
*[[Point group]]
*[[Point groups in two dimensions]]
*[[Point groups in three dimensions]]
*[[Fixed points of isometry groups in Euclidean space]]

==References==
{{reflist}}

[[Category:Metric geometry]]