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<title>Isometry group</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Isometry group</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>isometry group</b> of a <a href="Metric_space" title="Metric space">metric space</a> is the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of all <a href="Bijective" class="mw-redirect" title="Bijective">bijective</a> <a href="Isometry" title="Isometry">isometries</a> (that is, bijective, <a href="Distance-preserving_map" class="mw-redirect" title="Distance-preserving map">distance-preserving maps</a>) from the metric space onto itself, with the <a href="Function_composition" title="Function composition">function composition</a> as <a href="Group_(mathematics)" title="Group (mathematics)">group</a> operation.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Its <a href="Identity_element" title="Identity element">identity element</a> is the <a href="Identity_function" title="Identity function">identity function</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The elements of the isometry group are sometimes called <a href="Motion_(geometry)" title="Motion (geometry)">motions</a> of the space.

</p><p>Every isometry group of a metric space is a <a href="Subgroup" title="Subgroup">subgroup</a> of isometries. It represents in most cases a possible set of <a href="Symmetry" title="Symmetry">symmetries</a> of objects/figures in the space, or functions defined on the space. See <a href="Symmetry_group" title="Symmetry group">symmetry group</a>.

</p><p>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 <a href="Discrete_set" class="mw-redirect" title="Discrete set">discrete set</a>.

</p><p>In <a href="Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a> the metric is replaced with an <a href="Isotropic_quadratic_form" title="Isotropic quadratic form">isotropic quadratic form</a>; 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.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>The isometry group of the <a href="Linear_subspace" title="Linear subspace">subspace</a> of a <a href="Metric_space" title="Metric space">metric space</a> consisting of the points of a <a href="Triangle#Types_of_triangle" title="Triangle">scalene triangle</a> is the <a href="Trivial_group" title="Trivial group">trivial group</a>. A similar space for an <a href="Isosceles_triangle" title="Isosceles triangle">isosceles triangle</a> is the <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of <a href="Order_(group_theory)" title="Order (group theory)">order</a> two, C<sub>2</sub>. A similar space for an <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> is D<sub>3</sub>, the <a href="Dihedral_group_of_order_6" title="Dihedral group of order 6">dihedral group of order 6</a>.</li>

<li>The isometry group of a two-dimensional <a href="Sphere" title="Sphere">sphere</a> is the <a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(3).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul>

<ul><li>The isometry group of the <i>n</i>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> is the <a href="Euclidean_group" title="Euclidean group">Euclidean group</a> E(<i>n</i>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li>

<li>The isometry group of the <a href="Poincar%C3%A9_disc_model" class="mw-redirect" title="Poincaré disc model">Poincaré disc model</a> of the <a href="Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a> is the projective special unitary group <a href="Projective_special_unitary_group" class="mw-redirect" title="Projective special unitary group">PSU(1,1)</a>.</li>

<li>The isometry group of the <a href="Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a> of the hyperbolic plane is <a href="PSL(2%2CR)" class="mw-redirect" title="PSL(2,R)">PSL(2,R)</a>.</li>

<li>The isometry group of <a href="Minkowski_space" title="Minkowski space">Minkowski space</a> is the <a href="Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<ul><li><a href="Riemannian_symmetric_space" class="mw-redirect" title="Riemannian symmetric space">Riemannian symmetric spaces</a> are important cases where the isometry group is a <a href="Lie_group" title="Lie group">Lie group</a>.</li></ul>

<ul><li><a href="Point_group" title="Point group">Point group</a></li>

<li><a href="Point_groups_in_two_dimensions" title="Point groups in two dimensions">Point groups in two dimensions</a></li>

<li><a href="Point_groups_in_three_dimensions" title="Point groups in three dimensions">Point groups in three dimensions</a></li>

<li><a href="Fixed_points_of_isometry_groups_in_Euclidean_space" title="Fixed points of isometry groups in Euclidean space">Fixed points of isometry groups in Euclidean space</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFOlver1999" class="citation cs2"><a href="Peter_J._Olver" title="Peter J. Olver">Olver, Peter J.</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1GlHYhNRAqEC&pg=PA53"><i>Classical invariant theory</i></a>, London Mathematical Society Student Texts, vol. 44, Cambridge: Cambridge University Press, p. 53, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511623660">10.1017/CBO9780511623660</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-55821-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1694364">1694364</a></cite>.</span>

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