Orthogonal group - Wikipedia

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class="vector-toc-text"> 2.4 Symmetry group of spheres </div> </a> <ul id="toc-Symmetry_group_of_spheres-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Group_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Group_structure"> <div class="vector-toc-text"> 3 Group structure </div> </a> <button aria-controls="toc-Group_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Group structure subsection </button> <ul id="toc-Group_structure-sublist" class="vector-toc-list"> <li id="toc-As_algebraic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_algebraic_groups"> <div class="vector-toc-text"> 3.1 As algebraic groups </div> </a> <ul id="toc-As_algebraic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximal_tori_and_Weyl_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximal_tori_and_Weyl_groups"> <div class="vector-toc-text"> 3.2 Maximal tori and Weyl groups </div> </a> <ul id="toc-Maximal_tori_and_Weyl_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> 4 Topology </div> </a> <button aria-controls="toc-Topology-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Topology subsection </button> <ul id="toc-Topology-sublist" class="vector-toc-list"> <li id="toc-Low-dimensional_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Low-dimensional_topology"> <div class="vector-toc-text"> 4.1 Low-dimensional topology </div> </a> <ul id="toc-Low-dimensional_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fundamental_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamental_group"> <div class="vector-toc-text"> 4.2 Fundamental group </div> </a> <ul id="toc-Fundamental_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homotopy_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homotopy_groups"> <div class="vector-toc-text"> 4.3 Homotopy groups </div> </a> <ul id="toc-Homotopy_groups-sublist" class="vector-toc-list"> <li id="toc-Relation_to_KO-theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relation_to_KO-theory"> <div class="vector-toc-text"> 4.3.1 Relation to KO-theory </div> </a> <ul id="toc-Relation_to_KO-theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation_and_interpretation_of_homotopy_groups" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Computation_and_interpretation_of_homotopy_groups"> <div class="vector-toc-text"> 4.3.2 Computation and interpretation of homotopy groups </div> </a> <ul id="toc-Computation_and_interpretation_of_homotopy_groups-sublist" class="vector-toc-list"> <li id="toc-Low-dimensional_groups" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Low-dimensional_groups"> <div class="vector-toc-text"> 4.3.2.1 Low-dimensional groups </div> </a> <ul id="toc-Low-dimensional_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_groups" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Lie_groups"> <div class="vector-toc-text"> 4.3.2.2 Lie groups </div> </a> <ul id="toc-Lie_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_bundles" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Vector_bundles"> <div class="vector-toc-text"> 4.3.2.3 Vector bundles </div> </a> <ul id="toc-Vector_bundles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loop_spaces" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Loop_spaces"> <div class="vector-toc-text"> 4.3.2.4 Loop spaces </div> </a> <ul id="toc-Loop_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interpretation_of_homotopy_groups" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Interpretation_of_homotopy_groups"> <div class="vector-toc-text"> 4.3.3 Interpretation of homotopy groups </div> </a> <ul id="toc-Interpretation_of_homotopy_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Whitehead_tower" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Whitehead_tower"> <div class="vector-toc-text"> 4.3.4 Whitehead tower </div> </a> <ul id="toc-Whitehead_tower-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Of_indefinite_quadratic_form_over_the_reals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Of_indefinite_quadratic_form_over_the_reals"> <div class="vector-toc-text"> 5 Of indefinite quadratic form over the reals </div> </a> <ul id="toc-Of_indefinite_quadratic_form_over_the_reals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Of_complex_quadratic_forms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Of_complex_quadratic_forms"> <div class="vector-toc-text"> 6 Of complex quadratic forms </div> </a> <ul id="toc-Of_complex_quadratic_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Over_finite_fields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Over_finite_fields"> <div class="vector-toc-text"> 7 Over finite fields </div> </a> <button aria-controls="toc-Over_finite_fields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Over finite fields subsection </button> <ul id="toc-Over_finite_fields-sublist" class="vector-toc-list"> <li id="toc-Characteristic_different_from_two" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characteristic_different_from_two"> <div class="vector-toc-text"> 7.1 Characteristic different from two </div> </a> <ul id="toc-Characteristic_different_from_two-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dickson_invariant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dickson_invariant"> <div class="vector-toc-text"> 7.2 Dickson invariant </div> </a> <ul id="toc-Dickson_invariant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthogonal_groups_of_characteristic_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthogonal_groups_of_characteristic_2"> <div class="vector-toc-text"> 7.3 Orthogonal groups of characteristic 2 </div> </a> <ul id="toc-Orthogonal_groups_of_characteristic_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_spinor_norm" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_spinor_norm"> <div class="vector-toc-text"> 8 The spinor norm </div> </a> <ul id="toc-The_spinor_norm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Galois_cohomology_and_orthogonal_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Galois_cohomology_and_orthogonal_groups"> <div class="vector-toc-text"> 9 Galois cohomology and orthogonal groups </div> </a> <ul id="toc-Galois_cohomology_and_orthogonal_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_algebra" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lie_algebra"> <div class="vector-toc-text"> 10 Lie algebra </div> </a> <ul id="toc-Lie_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_groups"> <div class="vector-toc-text"> 11 Related groups </div> </a> <button aria-controls="toc-Related_groups-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Related groups subsection </button> <ul id="toc-Related_groups-sublist" class="vector-toc-list"> <li id="toc-Lie_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_subgroups"> <div class="vector-toc-text"> 11.1 Lie subgroups </div> </a> <ul id="toc-Lie_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_supergroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_supergroups"> <div class="vector-toc-text"> 11.2 Lie supergroups </div> </a> <ul id="toc-Lie_supergroups-sublist" class="vector-toc-list"> <li id="toc-Conformal_group" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conformal_group"> <div class="vector-toc-text"> 11.2.1 Conformal group </div> </a> <ul id="toc-Conformal_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Discrete_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_subgroups"> <div class="vector-toc-text"> 11.3 Discrete subgroups </div> </a> <ul id="toc-Discrete_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covering_and_quotient_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Covering_and_quotient_groups"> <div class="vector-toc-text"> 11.4 Covering and quotient groups </div> </a> <ul id="toc-Covering_and_quotient_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Principal_homogeneous_space:_Stiefel_manifold" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Principal_homogeneous_space:_Stiefel_manifold"> <div class="vector-toc-text"> 12 Principal homogeneous space: Stiefel manifold </div> </a> <ul id="toc-Principal_homogeneous_space:_Stiefel_manifold-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 13 See also </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle See also subsection </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Specific_transforms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific_transforms"> <div class="vector-toc-text"> 13.1 Specific transforms </div> </a> <ul id="toc-Specific_transforms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific_groups"> <div class="vector-toc-text"> 13.2 Specific groups </div> </a> <ul id="toc-Specific_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_groups_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_groups_2"> <div class="vector-toc-text"> 13.3 Related groups </div> </a> <ul id="toc-Related_groups_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lists_of_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lists_of_groups"> <div class="vector-toc-text"> 13.4 Lists of groups </div> </a> <ul id="toc-Lists_of_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representation_theory"> <div class="vector-toc-text"> 13.5 Representation theory </div> </a> <ul id="toc-Representation_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 14 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 15 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 16 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 17 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Orthogonal group</h1> <div 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Available in 20 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-20" aria-hidden="true" > 20 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D9%85%D8%AA%D8%B9%D8%A7%D9%85%D8%AF%D8%A9" title="زمرة متعامدة – Arabic" lang="ar" hreflang="ar" data-title="زمرة متعامدة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%80%D1%82%D0%B0%D0%B3%D0%B0%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Артаганальная група – Belarusian" lang="be" hreflang="be" data-title="Артаганальная група" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_ortogonal" title="Grup ortogonal – Catalan" lang="ca" hreflang="ca" data-title="Grup ortogonal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ortogon%C3%A1ln%C3%AD_grupa" title="Ortogonální grupa – Czech" lang="cs" hreflang="cs" data-title="Ortogonální grupa" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Orthogonale_Gruppe" title="Orthogonale Gruppe – German" lang="de" hreflang="de" data-title="Orthogonale Gruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_ortogonal" title="Grupo ortogonal – Spanish" lang="es" hreflang="es" data-title="Grupo ortogonal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_orthogonal" title="Groupe orthogonal – French" lang="fr" hreflang="fr" data-title="Groupe orthogonal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Possan_cair-uillinagh" title="Possan cair-uillinagh – Manx" lang="gv" hreflang="gv" data-title="Possan cair-uillinagh" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target">Gaelg</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%81%EA%B5%90%EA%B5%B0" title="직교군 – Korean" lang="ko" hreflang="ko" data-title="직교군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_ortogonale" title="Gruppo ortogonale – Italian" lang="it" hreflang="it" data-title="Gruppo ortogonale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Orthogonale_groep" title="Orthogonale groep – Dutch" lang="nl" hreflang="nl" data-title="Orthogonale groep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%9B%B4%E4%BA%A4%E7%BE%A4" title="直交群 – Japanese" lang="ja" hreflang="ja" data-title="直交群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_ortogonal" title="Grupo ortogonal – Portuguese" lang="pt" hreflang="pt" data-title="Grupo ortogonal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Grup_ortogonal" title="Grup ortogonal – Romanian" lang="ro" hreflang="ro" data-title="Grup ortogonal" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Ортогональная группа – Russian" lang="ru" hreflang="ru" data-title="Ортогональная группа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ortogonalgrupp" title="Ortogonalgrupp – Swedish" lang="sv" hreflang="sv" data-title="Ortogonalgrupp" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a 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href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Zn</li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> Sn</li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> An</li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> Dn</li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group">p-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">)</li><li>SL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(n)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(n)</li></ul> <ul><li><a class="mw-selflink selflink">Orthogonal</a> O(n)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(n)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(n)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(n)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(n)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(n)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E6</a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E7</a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E8</a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the orthogonal group in dimension n, denoted O(n), is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Isometry" title="Isometry">distance-preserving transformations</a> of a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> of dimension n that preserve a fixed point, where the group operation is given by <a href="/wiki/Function_composition" title="Function composition">composing</a> transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>. Equivalently, it is the group of n × n <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a>, where the group operation is given by <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> (an orthogonal matrix is a <a href="/wiki/Real_matrix" class="mw-redirect" title="Real matrix">real matrix</a> whose <a href="/wiki/Invertible_matrix" title="Invertible matrix">inverse</a> equals its <a href="/wiki/Transpose" title="Transpose">transpose</a>). The orthogonal group is an <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic group</a> and a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. It is <a href="/wiki/Compact_group" title="Compact group">compact</a>. The orthogonal group in dimension n has two <a href="/wiki/Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected components</a>. The one that contains the <a href="/wiki/Identity_element" title="Identity element">identity element</a> is a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of <a href="/wiki/Determinant" title="Determinant">determinant</a> 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see <a href="/wiki/SO(2)" class="mw-redirect" title="SO(2)">SO(2)</a>, <a href="/wiki/SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a> and <a href="/wiki/SO(4)" class="mw-redirect" title="SO(4)">SO(4)</a>. The other component consists of all orthogonal matrices of determinant −1. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component. By extension, for any field F, an n × n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n × n orthogonal matrices form a subgroup, denoted O(n, F), of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> GL(n, F); that is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>Q</mi> <mo>∈</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mo>=</mo> <mi>Q</mi> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>I</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ae53f9baa7adbce56646d3dcd268bad973eb33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.927ex; height:3.343ex;" alt="{\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}"> More generally, given a non-degenerate <a href="/wiki/Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a> or <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a><a href="#cite_note-1">[1]</a> on a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, the orthogonal group of the form is the group of invertible <a href="/wiki/Linear_map" title="Linear map">linear maps</a> that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the <a href="/wiki/Dot_product" title="Dot product">dot product</a>, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a>, since the condition of preserving a form can be expressed as an equality of matrices. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Name">Name</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=1" title="Edit section: Name">edit</a>]</div> The name of "orthogonal group" originates from the following characterization of its elements. Given a <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector space</a> E of dimension n, the elements of the orthogonal group O(n) are, <a href="/wiki/Up_to" title="Up to">up to</a> a <a href="/wiki/Uniform_scaling" class="mw-redirect" title="Uniform scaling">uniform scaling</a> (<a href="/wiki/Homothecy" class="mw-redirect" title="Homothecy">homothecy</a>), the <a href="/wiki/Linear_map" title="Linear map">linear maps</a> from E to E that map <a href="/wiki/Orthogonal_vector" class="mw-redirect" title="Orthogonal vector">orthogonal vectors</a> to orthogonal vectors. <div class="mw-heading mw-heading2"><h2 id="In_Euclidean_geometry">In Euclidean geometry</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=2" title="Edit section: In Euclidean geometry">edit</a>]</div> The orthogonal O(n) is the subgroup of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> GL(n, R), consisting of all <a href="/wiki/Endomorphisms" class="mw-redirect" title="Endomorphisms">endomorphisms</a> that preserve the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>; that is, endomorphisms g such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|g(x)\|=\|x\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|g(x)\|=\|x\|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952423c0b02e41f60653e7f982c7968dd2cbfc51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.98ex; height:2.843ex;" alt="{\displaystyle \|g(x)\|=\|x\|.}"> Let E(n) be the group of the <a href="/wiki/Euclidean_isometry" class="mw-redirect" title="Euclidean isometry">Euclidean isometries</a> of a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> S of dimension n. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a>. The <a href="/wiki/Stabilizer_subgroup" class="mw-redirect" title="Stabilizer subgroup">stabilizer subgroup</a> of a point x ∈ S is the subgroup of the elements g ∈ E(n) such that g(x) = x. This stabilizer is (or, more exactly, is isomorphic to) O(n), since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space. There is a natural <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> p from E(n) to O(n), which is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(g)(y-x)=g(y)-g(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(g)(y-x)=g(y)-g(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296ebaf47306f2d9c23dfae3bb1e312ff3e08aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:26.24ex; height:2.843ex;" alt="{\displaystyle p(g)(y-x)=g(y)-g(x),}"></dd></dl> where, as usual, the subtraction of two points denotes the <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by g (for details, see <a href="/wiki/Affine_space#Subtraction_and_Weyl's_axioms" title="Affine space">Affine space § Subtraction and Weyl's axioms</a>). The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of p is the vector space of the translations. So, the translations form a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of E(n), the stabilizers of two points are <a href="/wiki/Conjugate_subgroup" class="mw-redirect" title="Conjugate subgroup">conjugate</a> under the action of the translations, and all stabilizers are isomorphic to O(n). Moreover, the Euclidean group is a <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of O(n) and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of O(n). <div class="mw-heading mw-heading3"><h3 id="Special_orthogonal_group">Special orthogonal group</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=3" title="Edit section: Special orthogonal group">edit</a>]</div> By choosing an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of <a href="/wiki/Orthogonal_matrices" class="mw-redirect" title="Orthogonal matrices">orthogonal matrices</a>, which are the matrices such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QQ^{\mathsf {T}}=I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QQ^{\mathsf {T}}=I.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a735e54898214aab10ac4fb8eebcd93ef54b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.945ex; height:3.009ex;" alt="{\displaystyle QQ^{\mathsf {T}}=I.}"></dd></dl> It follows from this equation that the square of the <a href="/wiki/Determinant" title="Determinant">determinant</a> of Q equals 1, and thus the determinant of Q is either 1 or −1. The orthogonal matrices with determinant 1 form a subgroup called the special orthogonal group, denoted SO(n), consisting of all <a href="/wiki/Euclidean_group#Direct_and_indirect_isometries" title="Euclidean group">direct isometries</a> of O(n), which are those that preserve the <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation</a> of the space. SO(n) is a normal subgroup of O(n), as being the <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of the determinant, which is a group homomorphism whose image is the multiplicative group {−1, +1}. This implies that the orthogonal group is an internal <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of SO(n) and any subgroup formed with the identity and a <a href="/wiki/Reflection_(geometry)" class="mw-redirect" title="Reflection (geometry)">reflection</a>. The group with two elements {±I} (where I is the identity matrix) is a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> and even a <a href="/wiki/Characteristic_subgroup" title="Characteristic subgroup">characteristic subgroup</a> of O(n), and, if n is even, also of SO(n). If n is odd, O(n) is the internal <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of SO(n) and {±I}. The group SO(2) is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> (whereas SO(n) is not abelian when n > 2). Its finite subgroups are the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> Ck of <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">k-fold rotations</a>, for every positive integer k. All these groups are normal subgroups of O(2) and SO(2). <div class="mw-heading mw-heading3"><h3 id="Canonical_form">Canonical form</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=4" title="Edit section: Canonical form">edit</a>]</div> For any element of O(n) there is an orthogonal basis, where its matrix has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>±</mo> <mn>1</mn> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>±</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f62c045098cb32ab3ee373bdedc48c1d034de5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:35.184ex; height:22.509ex;" alt="{\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}"></dd></dl> where the matrices R1, ..., Rk are 2-by-2 rotation matrices, that is matrices of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511c75d021c31bba852462b067bfad41086f3052" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.211ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}},}"></dd></dl> with a2 + b2 = 1. This results from the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> by regrouping <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> that are <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to 1. The element belongs to SO(n) if and only if there are an even number of −1 on the diagonal. A pair of eigenvalues −1 can be identified with a rotation by π and a pair of eigenvalues +1 can be identified with a rotation by 0. The special case of n = 3 is known as <a href="/wiki/Euler%27s_rotation_theorem" title="Euler's rotation theorem">Euler's rotation theorem</a>, which asserts that every (non-identity) element of SO(3) is a <a href="/wiki/Rotation" title="Rotation">rotation</a> about a unique axis–angle pair. <div class="mw-heading mw-heading3"><h3 id="Reflections">Reflections</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=5" title="Edit section: Reflections">edit</a>]</div> <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">Reflections</a> are the elements of O(n) whose canonical form is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}-1&0\\0&I\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-1&0\\0&I\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e22808665f9d482793ff6a882a8ac0fd56c0396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.318ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}-1&0\\0&I\end{bmatrix}},}"></dd></dl> where I is the (n − 1) × (n − 1) identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a> with respect to a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a>. In dimension two, <a href="/wiki/Rotations_and_reflections_in_two_dimensions" title="Rotations and reflections in two dimensions">every rotation can be decomposed into a product of two reflections</a>. More precisely, a rotation of angle θ is the product of two reflections whose axes form an angle of θ / 2. A product of up to n elementary reflections always suffices to generate any element of O(n). This results immediately from the above canonical form and the case of dimension two. The <a href="/wiki/Cartan%E2%80%93Dieudonn%C3%A9_theorem" title="Cartan–Dieudonné theorem">Cartan–Dieudonné theorem</a> is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two. The <a href="/wiki/Reflection_through_the_origin" class="mw-redirect" title="Reflection through the origin">reflection through the origin</a> (the map v ↦ −v) is an example of an element of O(n) that is not a product of fewer than n reflections. <div class="mw-heading mw-heading3"><h3 id="Symmetry_group_of_spheres">Symmetry group of spheres</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=6" title="Edit section: Symmetry group of spheres">edit</a>]</div> The orthogonal group O(n) is the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of the <a href="/wiki/N-sphere" title="N-sphere">(n − 1)-sphere</a> (for n = 3, this is just the <a href="/wiki/Sphere" title="Sphere">sphere</a>) and all objects with spherical symmetry, if the origin is chosen at the center. The <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of a <a href="/wiki/Circle" title="Circle">circle</a> is O(2). The orientation-preserving subgroup SO(2) is isomorphic (as a real Lie group) to the <a href="/wiki/Circle_group" title="Circle group">circle group</a>, also known as <a href="/wiki/Unitary_group" title="Unitary group">U</a>(1), the multiplicative group of the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> of absolute value equal to one. This isomorphism sends the complex number exp(φ i) = cos(φ) + i sin(φ) of <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> 1 to the special orthogonal matrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992b0590e850c4daf909b5de5ebde221f5a8d640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.997ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{bmatrix}}.}"></dd></dl> In higher dimension, O(n) has a more complicated structure (in particular, it is no longer commutative). The <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> structures of the n-sphere and O(n) are strongly correlated, and this correlation is widely used for studying both <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. <div class="mw-heading mw-heading2"><h2 id="Group_structure">Group structure</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=7" title="Edit section: Group structure">edit</a>]</div> The groups O(n) and SO(n) are real <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> of <a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">dimension</a> n(n − 1) / 2. The group O(n) has two <a href="/wiki/Connected_space" title="Connected space">connected components</a>, with SO(n) being the <a href="/wiki/Identity_component" title="Identity component">identity component</a>, that is, the connected component containing the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. <div class="mw-heading mw-heading3"><h3 id="As_algebraic_groups">As algebraic groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=8" title="Edit section: As algebraic groups">edit</a>]</div> The orthogonal group O(n) can be identified with the group of the matrices A such that ATA = I. Since both members of this equation are <a href="/wiki/Symmetric_matrices" class="mw-redirect" title="Symmetric matrices">symmetric matrices</a>, this provides n(n + 1) / 2 equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix. This proves that O(n) is an <a href="/wiki/Algebraic_set" class="mw-redirect" title="Algebraic set">algebraic set</a>. Moreover, it can be proved[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] that its dimension is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n(n-1)}{2}}=n^{2}-{\frac {n(n+1)}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n(n-1)}{2}}=n^{2}-{\frac {n(n+1)}{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec5ae8404e91f4515111df65cde6cc9fc8219e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.91ex; height:5.676ex;" alt="{\displaystyle {\frac {n(n-1)}{2}}=n^{2}-{\frac {n(n+1)}{2}},}"></dd></dl> which implies that O(n) is a <a href="/wiki/Complete_intersection" title="Complete intersection">complete intersection</a>. This implies that all its <a href="/wiki/Irreducible_component" title="Irreducible component">irreducible components</a> have the same dimension, and that it has no <a href="/wiki/Embedded_prime" class="mw-redirect" title="Embedded prime">embedded component</a>. In fact, O(n) has two irreducible components, that are distinguished by the sign of the determinant (that is det(A) = 1 or det(A) = −1). Both are <a href="/wiki/Singular_point_of_an_algebraic_variety" title="Singular point of an algebraic variety">nonsingular algebraic varieties</a> of the same dimension n(n − 1) / 2. The component with det(A) = 1 is SO(n). <div class="mw-heading mw-heading3"><h3 id="Maximal_tori_and_Weyl_groups">Maximal tori and Weyl groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=9" title="Edit section: Maximal tori and Weyl groups">edit</a>]</div> A <a href="/wiki/Maximal_torus" title="Maximal torus">maximal torus</a> in a compact <a href="/wiki/Lie_group" title="Lie group">Lie group</a> G is a maximal subgroup among those that are isomorphic to Tk for some k, where T = SO(2) is the standard one-dimensional torus.<a href="#cite_note-2">[2]</a> In O(2n) and SO(2n), for every maximal torus, there is a basis on which the torus consists of the <a href="/wiki/Block_matrix#Block_diagonal_matrices" title="Block matrix">block-diagonal matrices</a> of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd /> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd5af8a6de711289e2e3bbcb12d2e79fa1ccd61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:17.924ex; height:11.009ex;" alt="{\displaystyle {\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},}"></dd></dl> where each Rj belongs to SO(2). In O(2n + 1) and SO(2n + 1), the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal. The <a href="/wiki/Maximal_torus#Weyl_group" title="Maximal torus">Weyl group</a> of SO(2n + 1) is the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\pm 1\}^{n}\rtimes S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋊</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\pm 1\}^{n}\rtimes S_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df4add7563f89713147119153adb1d8fcfcfc10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.998ex; height:2.843ex;" alt="{\displaystyle \{\pm 1\}^{n}\rtimes S_{n}}"> of a normal <a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">elementary abelian</a> <a href="/wiki/P-group" title="P-group">2-subgroup</a> and a <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a>, where the nontrivial element of each {±1} factor of {±1}n acts on the corresponding circle factor of T × {1} by <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">inversion</a>, and the symmetric group Sn acts on both {±1}n and T × {1} by permuting factors. The elements of the Weyl group are represented by matrices in O(2n) × {±1}. The Sn factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The {±1}n component is represented by block-diagonal matrices with 2-by-2 blocks either <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b65ddcddf406dd22917e7042e9277bea89bbcba2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.074ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},}"></dd></dl> with the last component ±1 chosen to make the determinant 1. The Weyl group of SO(2n) is the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⋊</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋊</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f549666d03f38b8263f896f657e7d9ab17bbd121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.83ex; height:2.843ex;" alt="{\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}}"> of that of SO(2n + 1), where Hn−1 < {±1}n is the <a href="/wiki/Kernel_(algebra)#Group_homomorphism" title="Kernel (algebra)">kernel</a> of the product homomorphism {±1}n → {±1} given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\varepsilon _{1},\ldots ,\varepsilon _{n}\right)\mapsto \varepsilon _{1}\cdots \varepsilon _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">↦</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\varepsilon _{1},\ldots ,\varepsilon _{n}\right)\mapsto \varepsilon _{1}\cdots \varepsilon _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d33a204db27e0304e2df9fff026ebb12004ea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.978ex; height:2.843ex;" alt="{\displaystyle \left(\varepsilon _{1},\ldots ,\varepsilon _{n}\right)\mapsto \varepsilon _{1}\cdots \varepsilon _{n}}">; that is, Hn−1 < {±1}n is the subgroup with an even number of minus signs. The Weyl group of SO(2n) is represented in SO(2n) by the preimages under the standard injection SO(2n) → SO(2n + 1) of the representatives for the Weyl group of SO(2n + 1). Those matrices with an odd number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc64578eb51f34058527d6bb06d6162a1908de83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.854ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}"> blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2n). <div class="mw-heading mw-heading2"><h2 id="Topology">Topology</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=10" title="Edit section: Topology">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Confusing plainlinks metadata ambox ambox-style ambox-confusing" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section may be <a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers. In particular, most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists essentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (no reference, no link to articles about the methods of computation that are used, no sketch of proofs. Please help <a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the section</a>. There might be a discussion about this on <a href="/wiki/Talk:Orthogonal_group" title="Talk:Orthogonal group">the talk page</a>. (November 2019) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section may be too technical for most readers to understand. Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Orthogonal_group&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details. (November 2019) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Low-dimensional_topology">Low-dimensional topology</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=11" title="Edit section: Low-dimensional topology">edit</a>]</div> The low-dimensional (real) orthogonal groups are familiar <a href="/wiki/Topological_space" title="Topological space">spaces</a>: <ul><li>O(1) = S0, a <a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">two</a>-point <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete space</a></li> <li>SO(1) = {1}</li> <li>SO(2) is <a href="/wiki/Circle" title="Circle">S1</a></li> <li><a href="/wiki/Rotation_group_SO(3)" class="mw-redirect" title="Rotation group SO(3)">SO(3)</a> is <a href="/wiki/Real_projective_space" title="Real projective space">RP3</a> <a href="#cite_note-3">[3]</a></li> <li>SO(4) is <a href="/wiki/Double_cover_(topology)" class="mw-redirect" title="Double cover (topology)">doubly covered</a> by <a href="/wiki/Special_unitary_group" title="Special unitary group">SU</a>(2) × SU(2) = <a href="/wiki/3-sphere" title="3-sphere">S3</a> × S3.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Fundamental_group">Fundamental group</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=12" title="Edit section: Fundamental group">edit</a>]</div> In terms of <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, for n > 2 the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of SO(n, R) is <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic of order 2</a>,<a href="#cite_note-4">[4]</a> and the <a href="/wiki/Spin_group" title="Spin group">spin group</a> Spin(n) is its <a href="/wiki/Universal_cover" class="mw-redirect" title="Universal cover">universal cover</a>. For n = 2 the fundamental group is <a href="/wiki/Infinite_cyclic" class="mw-redirect" title="Infinite cyclic">infinite cyclic</a> and the universal cover corresponds to the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> (the group Spin(2) is the unique connected <a href="/wiki/Double_cover_(topology)" class="mw-redirect" title="Double cover (topology)">2-fold cover</a>). <div class="mw-heading mw-heading3"><h3 id="Homotopy_groups">Homotopy groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=13" title="Edit section: Homotopy groups">edit</a>]</div> Generally, the <a href="/wiki/Homotopy_group" title="Homotopy group">homotopy groups</a> πk(O) of the real orthogonal group are related to <a href="/wiki/Homotopy_groups_of_spheres" title="Homotopy groups of spheres">homotopy groups of spheres</a>, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the <a href="/wiki/Direct_limit" title="Direct limit">direct limit</a> of the sequence of inclusions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (0)\subset \operatorname {O} (1)\subset \operatorname {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\operatorname {O} (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <mi>O</mi> <mo>=</mo> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (0)\subset \operatorname {O} (1)\subset \operatorname {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\operatorname {O} (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f35cb999e333d411ebcb01861c0ef4f35a68cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.501ex; height:7.009ex;" alt="{\displaystyle \operatorname {O} (0)\subset \operatorname {O} (1)\subset \operatorname {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\operatorname {O} (k)}"></dd></dl> Since the inclusions are all closed, hence <a href="/wiki/Cofibration" title="Cofibration">cofibrations</a>, this can also be interpreted as a union. On the other hand, <a href="/wiki/N-sphere" title="N-sphere">Sn</a> is a <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous space</a> for O(n + 1), and one has the following <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n)\to \operatorname {O} (n+1)\to S^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n)\to \operatorname {O} (n+1)\to S^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c44b697f4c79a2a516929cadf0b9b9760a823402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.642ex; height:2.843ex;" alt="{\displaystyle \operatorname {O} (n)\to \operatorname {O} (n+1)\to S^{n},}"></dd></dl> which can be understood as "The orthogonal group O(n + 1) acts <a href="/wiki/Transitive_action" class="mw-redirect" title="Transitive action">transitively</a> on the unit sphere Sn, and the <a href="/wiki/Stabilizer_(group_theory)" class="mw-redirect" title="Stabilizer (group theory)">stabilizer</a> of a point (thought of as a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a>) is the orthogonal group of the <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">perpendicular complement</a>, which is an orthogonal group one dimension lower." Thus the natural inclusion O(n) → O(n + 1) is <a href="/wiki/N-connected" class="mw-redirect" title="N-connected">(n − 1)-connected</a>, so the homotopy groups stabilize, and πk(O(n + 1)) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces. From <a href="/wiki/Bott_periodicity" class="mw-redirect" title="Bott periodicity">Bott periodicity</a> we obtain Ω8O ≅ O, therefore the homotopy groups of O are 8-fold periodic, meaning πk + 8(O) = πk(O), and one need only to list the lower 8 homotopy groups: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d227c990d9d42c451e302d967a653dad4e86c34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:15.405ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Relation_to_KO-theory">Relation to KO-theory</h4>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=14" title="Edit section: Relation to KO-theory">edit</a>]</div> Via the <a href="/wiki/Clutching_construction" title="Clutching construction">clutching construction</a>, homotopy groups of the stable space O are identified with stable vector bundles on spheres (<a href="/wiki/Up_to_isomorphism" class="mw-redirect" title="Up to isomorphism">up to isomorphism</a>), with a dimension shift of 1: πk(O) = πk + 1(BO). Setting KO = BO × Z = Ω−1O × Z (to make π0 fit into the periodicity), one obtains: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mi>O</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68ea77a8753cfc2a366ac2b1115f7779e49f4ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:17.471ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Computation_and_interpretation_of_homotopy_groups">Computation and interpretation of homotopy groups</h4>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=15" title="Edit section: Computation and interpretation of homotopy groups">edit</a>]</div> <div class="mw-heading mw-heading5"><h5 id="Low-dimensional_groups">Low-dimensional groups</h5>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=16" title="Edit section: Low-dimensional groups">edit</a>]</div> The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups. <ul><li>π0(O) = π0(O(1)) = Z / 2Z, from <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>-preserving/reversing (this class survives to O(2) and hence stably)</li> <li>π1(O) = π1(SO(3)) = Z / 2Z, which is <a href="/wiki/Spin_group" title="Spin group">spin</a> comes from SO(3) = RP3 = S3 / (Z / 2Z).</li> <li>π2(O) = π2(SO(3)) = 0, which surjects onto π2(SO(4)); this latter thus vanishes.</li></ul> <div class="mw-heading mw-heading5"><h5 id="Lie_groups">Lie groups</h5>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=17" title="Edit section: Lie groups">edit</a>]</div> From general facts about <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>, π2(G) always vanishes, and π3(G) is free (<a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian</a>). <div class="mw-heading mw-heading5"><h5 id="Vector_bundles">Vector bundles</h5>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=18" title="Edit section: Vector bundles">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Confusing plainlinks metadata ambox ambox-style ambox-confusing" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section may be <a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers. Please help <a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the section</a>. There might be a discussion about this on <a href="/wiki/Talk:Orthogonal_group" title="Talk:Orthogonal group">the talk page</a>. (January 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> π0(KO) is a <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> over S0, which consists of two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so π0(KO) = <a href="/wiki/Integers" class="mw-redirect" title="Integers">Z</a> is the <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">dimension</a>. <div class="mw-heading mw-heading5"><h5 id="Loop_spaces">Loop spaces</h5>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=19" title="Edit section: Loop spaces">edit</a>]</div> Using concrete descriptions of the loop spaces in <a href="/wiki/Bott_periodicity" class="mw-redirect" title="Bott periodicity">Bott periodicity</a>, one can interpret the higher homotopies of O in terms of simpler-to-analyze homotopies of lower order. Using π0, O and O/U have two components, KO = BO × Z and KSp = BSp × Z have <a href="/wiki/Countably_many" class="mw-redirect" title="Countably many">countably many</a> components, and the rest are connected. <div class="mw-heading mw-heading4"><h4 id="Interpretation_of_homotopy_groups">Interpretation of homotopy groups</h4>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=20" title="Edit section: Interpretation of homotopy groups">edit</a>]</div> In a nutshell:<a href="#cite_note-5">[5]</a> <ul><li>π0(KO) = Z is about <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">dimension</a></li> <li>π1(KO) = Z / 2Z is about <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a></li> <li>π2(KO) = Z / 2Z is about <a href="/wiki/Spin_group" title="Spin group">spin</a></li> <li>π4(KO) = Z is about <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a>.</li></ul> Let R be any of the four <a href="/wiki/Division_algebra" title="Division algebra">division algebras</a> R, C, <a href="/wiki/Quaternions" class="mw-redirect" title="Quaternions">H</a>, <a href="/wiki/Octonions" class="mw-redirect" title="Octonions">O</a>, and let LR be the <a href="/wiki/Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">tautological line bundle</a> over the <a href="/wiki/Projective_line" title="Projective line">projective line</a> RP1, and [LR] its class in K-theory. Noting that <a href="/wiki/Real_projective_line" title="Real projective line">RP1</a> = S1, <a href="/wiki/Riemann_sphere" title="Riemann sphere">CP1</a> = S2, HP1 = S4, OP1 = S8, these yield vector bundles over the corresponding spheres, and <ul><li>π1(KO) is generated by [LR]</li> <li>π2(KO) is generated by [LC]</li> <li>π4(KO) is generated by [LH]</li> <li>π8(KO) is generated by [LO]</li></ul> From the point of view of <a href="/wiki/Symplectic_geometry" title="Symplectic geometry">symplectic geometry</a>, π0(KO) ≅ π8(KO) = Z can be interpreted as the <a href="/wiki/Maslov_index" class="mw-redirect" title="Maslov index">Maslov index</a>, thinking of it as the fundamental group π1(U/O) of the stable <a href="/wiki/Lagrangian_Grassmannian" title="Lagrangian Grassmannian">Lagrangian Grassmannian</a> as U/O ≅ Ω7(KO), so π1(U/O) = π1+7(KO). <div class="mw-heading mw-heading4"><h4 id="Whitehead_tower">Whitehead tower</h4>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=21" title="Edit section: Whitehead tower">edit</a>]</div> The orthogonal group anchors a <a href="/wiki/Whitehead_tower" class="mw-redirect" title="Whitehead tower">Whitehead tower</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋯</mo> <mo stretchy="false">→</mo> <mi>Fivebrane</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>String</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Spin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1efe7d69c364462cd202a4954b727ccb33946b03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.574ex; height:2.843ex;" alt="{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}"></dd></dl> which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequences</a> starting with an <a href="/wiki/Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a> for the homotopy group to be removed. The first few entries in the tower are the <a href="/wiki/Spin_group" title="Spin group">spin group</a> and the <a href="/wiki/String_group" title="String group">string group</a>, and are preceded by the <a href="/wiki/Fivebrane_group" class="mw-redirect" title="Fivebrane group">fivebrane group</a>. The homotopy groups that are killed are in turn π0(O) to obtain SO from O, π1(O) to obtain Spin from SO, π3(O) to obtain String from Spin, and then π7(O) and so on to obtain the higher order <a href="/wiki/Brane" title="Brane">branes</a>. <div class="mw-heading mw-heading2"><h2 id="Of_indefinite_quadratic_form_over_the_reals">Of indefinite quadratic form over the reals</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=22" title="Edit section: Of indefinite quadratic form over the reals">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">Indefinite orthogonal group</a></div> Over the real numbers, <a href="/wiki/Nondegenerate_quadratic_form" class="mw-redirect" title="Nondegenerate quadratic form">nondegenerate quadratic forms</a> are classified by <a href="/wiki/Sylvester%27s_law_of_inertia" title="Sylvester's law of inertia">Sylvester's law of inertia</a>, which asserts that, on a vector space of dimension n, such a form can be written as the difference of a sum of p squares and a sum of q squares, with p + q = n. In other words, there is a basis on which the matrix of the quadratic form is a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>, with p entries equal to 1, and q entries equal to −1. The pair (p, q) called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix. The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted O(p, q). Moreover, as a quadratic form and its opposite have the same orthogonal group, one has O(p, q) = O(q, p). The standard orthogonal group is O(n) = O(n, 0) = O(0, n). So, in the remainder of this section, it is supposed that neither p nor q is zero. The subgroup of the matrices of determinant 1 in O(p, q) is denoted SO(p, q). The group O(p, q) has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted SO+(p, q). The group O(3, 1) is the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> that is fundamental in <a href="/wiki/Relativity_theory" class="mw-redirect" title="Relativity theory">relativity theory</a>. Here the 3 corresponds to space coordinates, and 1 corresponds to the time coordinate. <div class="mw-heading mw-heading2"><h2 id="Of_complex_quadratic_forms">Of complex quadratic forms</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=23" title="Edit section: Of complex quadratic forms">edit</a>]</div> Over the field C of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, every non-degenerate <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> in n variables is equivalent to x12 + ... + xn2. Thus, up to isomorphism, there is only one non-degenerate complex <a href="/wiki/Quadratic_space" class="mw-redirect" title="Quadratic space">quadratic space</a> of dimension n, and one associated orthogonal group, usually denoted O(n, C). It is the group of complex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix. As in the real case, O(n, C) has two connected components. The component of the identity consists of all matrices of determinant 1 in O(n, C); it is denoted SO(n, C). The groups O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1) / 2 over C (the dimension over R is twice that). For n ≥ 2, these groups are noncompact. As in the real case, SO(n, C) is not simply connected: For n > 2, the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of SO(n, C) is <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic of order 2</a>, whereas the fundamental group of SO(2, C) is Z. <div class="mw-heading mw-heading2"><h2 id="Over_finite_fields">Over finite fields</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=24" title="Edit section: Over finite fields">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Characteristic_different_from_two">Characteristic different from two</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=25" title="Edit section: Characteristic different from two">edit</a>]</div> Over a field of characteristic different from two, two <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> are equivalent if their matrices are <a href="/wiki/Congruent_matrices" class="mw-redirect" title="Congruent matrices">congruent</a>, that is if a change of basis transforms the matrix of the first form into the matrix of the second form. Two equivalent quadratic forms have clearly the same orthogonal group. The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension. More precisely, <a href="/wiki/Witt%27s_decomposition_theorem" class="mw-redirect" title="Witt's decomposition theorem">Witt's decomposition theorem</a> asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form Q can be decomposed as a direct sum of pairwise orthogonal subspaces <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⊕</mo> <mi>W</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa9861ad86784a9835e4135bad4c56417761e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.585ex; height:2.509ex;" alt="{\displaystyle V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,}"></dd></dl> where each Li is a <a href="/wiki/Hyperbolic_plane_(quadratic_forms)" class="mw-redirect" title="Hyperbolic plane (quadratic forms)">hyperbolic plane</a> (that is there is a basis such that the matrix of the restriction of Q to Li has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d86609df11c3e1ae8abb736606e7ee7873a147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.854ex; height:6.176ex;" alt="{\displaystyle \textstyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}">), and the restriction of Q to W is <a href="/wiki/Anisotropic_quadratic_form" class="mw-redirect" title="Anisotropic quadratic form">anisotropic</a> (that is, Q(w) ≠ 0 for every nonzero w in W). The <a href="/wiki/Chevalley%E2%80%93Warning_theorem" title="Chevalley–Warning theorem">Chevalley–Warning theorem</a> asserts that, over a <a href="/wiki/Finite_field" title="Finite field">finite field</a>, the dimension of W is at most two. If the dimension of V is odd, the dimension of W is thus equal to one, and its matrix is congruent either to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\begin{bmatrix}1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\begin{bmatrix}1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b8be060a298b7887a995201ab96eb6ba5c9d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.208ex; height:2.843ex;" alt="{\displaystyle \textstyle {\begin{bmatrix}1\end{bmatrix}}}"> or to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\begin{bmatrix}\varphi \end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\begin{bmatrix}\varphi \end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70ceaac59f869c20293e3c64ba0860b99373a91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.212ex; height:2.843ex;" alt="{\displaystyle \textstyle {\begin{bmatrix}\varphi \end{bmatrix}},}"> where 𝜑 is a non-square scalar. It results that there is only one orthogonal group that is denoted O(2n + 1, q), where q is the number of elements of the finite field (a power of an odd prime).<a href="#cite_note-Wil6975-6">[6]</a> If the dimension of W is two and −1 is not a square in the ground field (that is, if its number of elements q is congruent to 3 modulo 4), the matrix of the restriction of Q to W is congruent to either I or –I, where I is the 2×2 identity matrix. If the dimension of W is two and −1 is a square in the ground field (that is, if q is congruent to 1, modulo 4) the matrix of the restriction of Q to W is congruent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\begin{bmatrix}1&0\\0&\varphi \end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\begin{bmatrix}1&0\\0&\varphi \end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6cc0906d7fa6ac420c4df04f03c677d95b03a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.858ex; height:6.176ex;" alt="{\displaystyle \textstyle {\begin{bmatrix}1&0\\0&\varphi \end{bmatrix}},}"> φ is any non-square scalar. This implies that if the dimension of V is even, there are only two orthogonal groups, depending whether the dimension of W zero or two. They are denoted respectively O+(2n, q) and O−(2n, q).<a href="#cite_note-Wil6975-6">[6]</a> The orthogonal group Oε(2, q) is a <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> of order 2(q − ε), where ε = ±. <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style="">Proof For studying the orthogonal group of Oε(2, q), one can suppose that the matrix of the quadratic form is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\begin{bmatrix}1&0\\0&-\omega \end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>ω</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\begin{bmatrix}1&0\\0&-\omega \end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e015879eb1adeddae5845ed39f2dc1284cd54880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.529ex; height:6.176ex;" alt="{\displaystyle Q={\begin{bmatrix}1&0\\0&-\omega \end{bmatrix}},}"> because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69fc0502c8f960b4cd20db0ea2a2acd263b6137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.816ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}"> belongs to the orthogonal group if AQAT = Q, that is, a2 – ωb2 = 1, ac – ωbd = 0, and c2 – ωd2 = –ω. As a and b cannot be both zero (because of the first equation), the second equation implies the existence of ε in Fq, such that c = εωb and d = εa. Reporting these values in the third equation, and using the first equation, one gets that ε2 = 1, and thus the orthogonal group consists of the matrices <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a&b\\\varepsilon \omega b&\varepsilon a\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>ε</mi> <mi>ω</mi> <mi>b</mi> </mtd> <mtd> <mi>ε</mi> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&b\\\varepsilon \omega b&\varepsilon a\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e12363ccd7c6af8cdf6a2824ab9674c6ebb823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.016ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}a&b\\\varepsilon \omega b&\varepsilon a\end{bmatrix}},}"></dd></dl> where a2 – ωb2 = 1 and ε = ±1. Moreover, the determinant of the matrix is ε. For further studying the orthogonal group, it is convenient to introduce a square root α of ω. This square root belongs to Fq if the orthogonal group is O+(2, q), and to Fq2 otherwise. Setting x = a + αb, and y = a – αb, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=1,\qquad a={\frac {x+y}{2}}\qquad b={\frac {x-y}{2\alpha }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="2em" /> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="2em" /> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=1,\qquad a={\frac {x+y}{2}}\qquad b={\frac {x-y}{2\alpha }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b71cd2c0ce64b9c21e41f479faaa97428c14e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.465ex; height:5.176ex;" alt="{\displaystyle xy=1,\qquad a={\frac {x+y}{2}}\qquad b={\frac {x-y}{2\alpha }}.}"></dd></dl> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}={\begin{bmatrix}a_{1}&b_{1}\\\omega b_{1}&a_{1}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}={\begin{bmatrix}a_{1}&b_{1}\\\omega b_{1}&a_{1}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c53b9afc285442f179dcd498c6389ba578ba2ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.206ex; height:6.176ex;" alt="{\displaystyle A_{1}={\begin{bmatrix}a_{1}&b_{1}\\\omega b_{1}&a_{1}\end{bmatrix}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}={\begin{bmatrix}a_{2}&b_{2}\\\omega b_{2}&a_{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}={\begin{bmatrix}a_{2}&b_{2}\\\omega b_{2}&a_{2}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f041ef7e30b154b2d83a4b478e65397a660a8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.206ex; height:6.176ex;" alt="{\displaystyle A_{2}={\begin{bmatrix}a_{2}&b_{2}\\\omega b_{2}&a_{2}\end{bmatrix}}}"> are two matrices of determinant one in the orthogonal group then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}A_{2}={\begin{bmatrix}a_{1}a_{2}+\omega b_{1}b_{2}&a_{1}b_{2}+b_{1}a_{2}\\\omega b_{1}a_{2}+\omega a_{1}b_{2}&\omega b_{1}b_{2}+a_{1}a_{1}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>ω</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>ω</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi>ω</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}A_{2}={\begin{bmatrix}a_{1}a_{2}+\omega b_{1}b_{2}&a_{1}b_{2}+b_{1}a_{2}\\\omega b_{1}a_{2}+\omega a_{1}b_{2}&\omega b_{1}b_{2}+a_{1}a_{1}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7a6e40af7eeb198032e98f3f438f2ab37222d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.231ex; height:6.176ex;" alt="{\displaystyle A_{1}A_{2}={\begin{bmatrix}a_{1}a_{2}+\omega b_{1}b_{2}&a_{1}b_{2}+b_{1}a_{2}\\\omega b_{1}a_{2}+\omega a_{1}b_{2}&\omega b_{1}b_{2}+a_{1}a_{1}\end{bmatrix}}.}"></dd></dl> This is an orthogonal matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52f1ad649ff3517bbb619a87631b9e43603528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.849ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},}"> with a = a1a2 + ωb1b2, and b = a1b2 + b1a2. Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+\alpha b=(a_{1}+\alpha b_{1})(a_{2}+\alpha b_{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>α</mi> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>α</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>α</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+\alpha b=(a_{1}+\alpha b_{1})(a_{2}+\alpha b_{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4839eb7c1529ed06aaf88464becc35cfc4d372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.247ex; height:2.843ex;" alt="{\displaystyle a+\alpha b=(a_{1}+\alpha b_{1})(a_{2}+\alpha b_{2}).}"></dd></dl> It follows that the map (a, b) ↦ a + αb is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of Fq2. In the case of O+(2n, q), the image is the multiplicative group of Fq, which is a cyclic group of order q. In the case of O–(2n, q), the above x and y are <a href="/wiki/Conjugate_element_(field_theory)" title="Conjugate element (field theory)">conjugate</a>, and are therefore the image of each other by the <a href="/wiki/Frobenius_automorphism" class="mw-redirect" title="Frobenius automorphism">Frobenius automorphism</a>. This meant that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{-1}=x^{q},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{-1}=x^{q},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3a8f811958dbe12022b5fbd6546d16402419cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.98ex; height:3.009ex;" alt="{\displaystyle y=x^{-1}=x^{q},}"> and thus xq+1 = 1. For every such x one can reconstruct a corresponding orthogonal matrix. It follows that the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\mapsto a+\alpha b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>+</mo> <mi>α</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\mapsto a+\alpha b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1026d77d240cb4db96e49c317467ace881b4a5d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.24ex; height:2.843ex;" alt="{\displaystyle (a,b)\mapsto a+\alpha b}"> is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the (q + 1)-<a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>. This group is a cyclic group of order q + 1 which consists of the powers of gq−1, where g is a <a href="/wiki/Primitive_element_(finite_field)" title="Primitive element (finite field)">primitive element</a> of Fq2, For finishing the proof, it suffices to verify that the group all orthogonal matrices is not abelian, and is the semidirect product of the group {1, −1} and the group of orthogonal matrices of determinant one. The comparison of this proof with the real case may be illuminating. Here two group isomorphisms are involved: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {Z} /(q+1)\mathbf {Z} &\to T\\k&\mapsto g^{(q-1)k},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>T</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {Z} /(q+1)\mathbf {Z} &\to T\\k&\mapsto g^{(q-1)k},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d5b98c16f754b6d0cdd3a60839b16bf19d0ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.668ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {Z} /(q+1)\mathbf {Z} &\to T\\k&\mapsto g^{(q-1)k},\end{aligned}}}"></dd></dl> where g is a primitive element of Fq2 and T is the multiplicative group of the element of norm one in Fq2 ; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {T} &\to \operatorname {SO} ^{+}(2,\mathbf {F} _{q})\\x&\mapsto {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <msup> <mi>SO</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {T} &\to \operatorname {SO} ^{+}(2,\mathbf {F} _{q})\\x&\mapsto {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211ad31c1ee78b6de11fc7d25a6724e1c8abc98b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.873ex; margin-bottom: -0.299ex; width:17.513ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {T} &\to \operatorname {SO} ^{+}(2,\mathbf {F} _{q})\\x&\mapsto {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},\end{aligned}}}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {x+x^{-1}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {x+x^{-1}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a79c6e1698e687564710455884dd6044f04cee85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.997ex; height:5.676ex;" alt="{\displaystyle a={\frac {x+x^{-1}}{2}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\frac {x-x^{-1}}{2\alpha }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\frac {x-x^{-1}}{2\alpha }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af81284879c0cf71e95be9f4e79ddf867e63de5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.412ex; height:5.676ex;" alt="{\displaystyle b={\frac {x-x^{-1}}{2\alpha }}.}"> In the real case, the corresponding isomorphisms are: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {R} /2\pi \mathbf {R} &\to C\\\theta &\mapsto e^{i\theta },\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {R} /2\pi \mathbf {R} &\to C\\\theta &\mapsto e^{i\theta },\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d72e8f2a0a579a27edfbfd587e278bcd96b904d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.33ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {R} /2\pi \mathbf {R} &\to C\\\theta &\mapsto e^{i\theta },\end{aligned}}}"></dd></dl> where C is the circle of the complex numbers of norm one; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {C} &\to \operatorname {SO} (2,\mathbf {R} )\\x&\mapsto {\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {C} &\to \operatorname {SO} (2,\mathbf {R} )\\x&\mapsto {\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20dcbdbeba07f9295dbf6ade713f81a0ac29062a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.59ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {C} &\to \operatorname {SO} (2,\mathbf {R} )\\x&\mapsto {\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}},\end{aligned}}}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e128422fe183eff61ef1383fe0d8230f4f41b85c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.951ex; height:5.676ex;" alt="{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6604741a33de7f83484cf1813f624cf3114ae579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.342ex; height:5.676ex;" alt="{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}.}"> </div> When the characteristic is not two, the order of the orthogonal groups are<a href="#cite_note-7">[7]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\operatorname {O} (2n+1,q)\right|=2q^{n^{2}}\prod _{i=1}^{n}\left(q^{2i}-1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {O} (2n+1,q)\right|=2q^{n^{2}}\prod _{i=1}^{n}\left(q^{2i}-1\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1563000f033a2f96525e1b5a35817bb34a3a9133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.576ex; height:6.843ex;" alt="{\displaystyle \left|\operatorname {O} (2n+1,q)\right|=2q^{n^{2}}\prod _{i=1}^{n}\left(q^{2i}-1\right),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\operatorname {O} ^{+}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}-1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msup> <mi mathvariant="normal">O</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {O} ^{+}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}-1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322f19d4d7ade017568953f41e81056cc58c8355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.233ex; height:7.343ex;" alt="{\displaystyle \left|\operatorname {O} ^{+}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}-1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\operatorname {O} ^{-}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msup> <mi mathvariant="normal">O</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {O} ^{-}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38d27749cbff0b67d7d9cd79e68741185bae5cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.233ex; height:7.343ex;" alt="{\displaystyle \left|\operatorname {O} ^{-}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right).}"></dd></dl> In characteristic two, the formulas are the same, except that the factor 2 of |O(2n + 1, q)| must be removed. <div class="mw-heading mw-heading3"><h3 id="Dickson_invariant">Dickson invariant</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=26" title="Edit section: Dickson invariant">edit</a>]</div> For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z / 2Z (integers modulo 2), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise.<a href="#cite_note-Knus224-8">[8]</a> Algebraically, the Dickson invariant can be defined as D(f) = rank(I − f) modulo 2, where I is the identity (<a href="#CITEREFTaylor1992">Taylor 1992</a>, Theorem 11.43). Over fields that are not of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant. The special orthogonal group is the <a href="/wiki/Kernel_(matrix)" class="mw-redirect" title="Kernel (matrix)">kernel</a> of the Dickson invariant<a href="#cite_note-Knus224-8">[8]</a> and usually has index 2 in O(n, F ).<a href="#cite_note-9">[9]</a> When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n, F ) is commonly defined to be the elements of O(n, F ) with determinant 1. Each element in O(n, F ) has determinant ±1. Thus in characteristic 2, the determinant is always 1. The Dickson invariant can also be defined for <a href="/wiki/Clifford_group" title="Clifford group">Clifford groups</a> and <a href="/wiki/Pin_group" title="Pin group">pin groups</a> in a similar way (in all dimensions). <div class="mw-heading mw-heading3"><h3 id="Orthogonal_groups_of_characteristic_2">Orthogonal groups of characteristic 2</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=27" title="Edit section: Orthogonal groups of characteristic 2">edit</a>]</div> Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the hypoabelian groups, but this term is no longer used.) <ul><li>Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the <a href="/wiki/Witt_index" class="mw-redirect" title="Witt index">Witt index</a> is 2.<a href="#cite_note-10">[10]</a> A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v + B(v, u)/Q(u) · u where B is the bilinear form[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This seems like it is the associated polar form B′(x,y)=Q(x+y)−Q(x)−Q(y) (the name 'associated bilinear form' is used variously to mean B′ or B=B′/2). When expressed in the same terms (e.g. in terms of Q), the expression for a reflection is the same for all cases. (May 2020)">clarification needed</a>] and Q is the quadratic form associated to the orthogonal geometry. Compare this to the <a href="/wiki/Householder_reflection" class="mw-redirect" title="Householder reflection">Householder reflection</a> of odd characteristic or characteristic zero, which takes v to v − 2·B(v, u)/Q(u) · u.</li> <li>The <a href="/wiki/Center_of_a_group" class="mw-redirect" title="Center of a group">center</a> of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = −I.</li> <li>In odd dimensions 2n + 1 in characteristic 2, orthogonal groups over <a href="/wiki/Perfect_field" title="Perfect field">perfect fields</a> are the same as <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic groups</a> in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.</li> <li>In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.</li></ul> <div class="mw-heading mw-heading2"><h2 id="The_spinor_norm">The spinor norm</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=28" title="Edit section: The spinor norm">edit</a>]</div> The spinor norm is a homomorphism from an orthogonal group over a field F to the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> F× / (F×)2 (the <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a> of the field F <a href="/wiki/Up_to" title="Up to">up to</a> multiplication by <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> elements), that takes reflection in a vector of norm n to the image of n in F× / (F×)2.<a href="#cite_note-C178-11">[11]</a> For the usual orthogonal group over the reals, it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite. <div class="mw-heading mw-heading2"><h2 id="Galois_cohomology_and_orthogonal_groups">Galois cohomology and orthogonal groups</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=29" title="Edit section: Galois cohomology and orthogonal groups">edit</a>]</div> In the theory of <a href="/wiki/Galois_cohomology" title="Galois cohomology">Galois cohomology</a> of <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a>, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomenon is concerned. The first point is that <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> over a field can be identified as a Galois H1, or twisted forms (<a href="/wiki/Torsor" class="mw-redirect" title="Torsor">torsors</a>) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the <a href="/wiki/Determinant" title="Determinant">determinant</a>. The 'spin' name of the spinor norm can be explained by a connection to the <a href="/wiki/Spin_group" title="Spin group">spin group</a> (more accurately a <a href="/wiki/Pin_group" title="Pin group">pin group</a>). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a>). The spin covering of the orthogonal group provides a <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </msub> </mrow> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1975a2040c979f28a8a721a89af2d3efdbfa8cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.528ex; height:2.676ex;" alt="{\displaystyle 1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1}"></dd></dl> Here μ2 is the <a href="/wiki/Group_scheme_of_roots_of_unity" class="mw-redirect" title="Group scheme of roots of unity">algebraic group of square roots of 1</a>; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The <a href="/wiki/Connecting_homomorphism" class="mw-redirect" title="Connecting homomorphism">connecting homomorphism</a> from H0(OV), which is simply the group OV(F) of F-valued points, to H1(μ2) is essentially the spinor norm, because H1(μ2) is isomorphic to the multiplicative group of the field modulo squares. There is also the connecting homomorphism from H1 of the orthogonal group, to the H2 of the kernel of the spin covering. The cohomology is non-abelian so that this is as far as we can go, at least with the conventional definitions. <div class="mw-heading mw-heading2"><h2 id="Lie_algebra">Lie algebra</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=30" title="Edit section: Lie algebra">edit</a>]</div> The <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a> n × n matrices, with the Lie bracket [ , ] given by the <a href="/wiki/Commutator" title="Commutator">commutator</a>. One Lie algebra corresponds to both groups. It is often denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {o}}(n,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {o}}(n,F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f806a7f667e931ae01ee8b51653e62e32dc4b3b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.116ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {o}}(n,F)}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(n,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n,F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e919d7f02580764bd39f74021b78fb0b08c5e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:8.186ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(n,F)}">, and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the <a href="/wiki/Compact_real_form" class="mw-redirect" title="Compact real form">compact real forms</a> of two of the four families of <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebras</a>: in odd dimension Bk, where n = 2k + 1, while in even dimension Dr, where n = 2r. Since the group SO(n) is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ordinary representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups. (The projective representations of SO(n) are just linear representations of the universal cover, the <a href="/wiki/Spin_group" title="Spin group">spin group</a> Spin(n).) The latter are the so-called <a href="/wiki/Spin_representation" title="Spin representation">spin representation</a>, which are important in physics. More generally, given a vector space V (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form (⋅, ⋅), the special orthogonal Lie algebra consists of tracefree endomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> which are skew-symmetric for this form (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi A,B)+(A,\varphi B)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>φ</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi A,B)+(A,\varphi B)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbec464f358a8ccabf070e5fca37455b9f7ec30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.842ex; height:2.843ex;" alt="{\displaystyle (\varphi A,B)+(A,\varphi B)=0}">). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors Λ2V. The correspondence is given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\wedge w\mapsto (v,\cdot )w-(w,\cdot )v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∧</mo> <mi>w</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mi>w</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\wedge w\mapsto (v,\cdot )w-(w,\cdot )v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be8aec7ba14076418174342ee019e0b72555a49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.392ex; height:2.843ex;" alt="{\displaystyle v\wedge w\mapsto (v,\cdot )w-(w,\cdot )v}"></dd></dl> This description applies equally for the indefinite special orthogonal Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(p,q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee93eecf623e48878214a948efa289b4193dcc74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.29ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(p,q)}"> for symmetric bilinear forms with signature (p, q). Over real numbers, this characterization is used in interpreting the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. <div class="mw-heading mw-heading2"><h2 id="Related_groups">Related groups</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=31" title="Edit section: Related groups">edit</a>]</div> The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below. The inclusions O(n) ⊂ U(n) ⊂ USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in a <a href="/wiki/Bott_periodicity_theorem#Geometric_model_of_loop_spaces" title="Bott periodicity theorem">geometric proof of the Bott periodicity theorem</a>, and the corresponding quotient spaces are <a href="/wiki/Symmetric_space" title="Symmetric space">symmetric spaces</a> of independent interest – for example, U(n)/O(n) is the <a href="/wiki/Lagrangian_Grassmannian" title="Lagrangian Grassmannian">Lagrangian Grassmannian</a>. <div class="mw-heading mw-heading3"><h3 id="Lie_subgroups">Lie subgroups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=32" title="Edit section: Lie subgroups">edit</a>]</div> In physics, particularly in the areas of <a href="/wiki/Kaluza%E2%80%93Klein" class="mw-redirect" title="Kaluza–Klein">Kaluza–Klein</a> compactification, it is important to find out the subgroups of the orthogonal group. The main ones are: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (n)\supset \mathrm {O} (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (n)\supset \mathrm {O} (n-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9908269df75e85beb70f2339afff4e8d35f7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.126ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (n)\supset \mathrm {O} (n-1)}"> – preserve an axis</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8694d24d7dd507d7952639906eb5665fb0845c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.558ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)}"> – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see <a href="/wiki/Unitary_group#2-out-of-3_property" title="Unitary group">2-out-of-3 property</a>; SU(n) also preserves a complex orientation.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (2n)\supset \mathrm {USp} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (2n)\supset \mathrm {USp} (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0a88bab7096722760d6ea131b5448c8ee941e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.805ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (2n)\supset \mathrm {USp} (n)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (7)\supset \mathrm {G} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>⊃</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (7)\supset \mathrm {G} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9c842f802294d1d7dca69954d1cfed3b0f4617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.757ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (7)\supset \mathrm {G} _{2}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Lie_supergroups">Lie supergroups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=33" title="Edit section: Lie supergroups">edit</a>]</div> The orthogonal group O(n) is also an important subgroup of various Lie groups: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {U} (n)&\supset \mathrm {O} (n)\\\mathrm {USp} (2n)&\supset \mathrm {O} (n)\\\mathrm {G} _{2}&\supset \mathrm {O} (3)\\\mathrm {F} _{4}&\supset \mathrm {O} (9)\\\mathrm {E} _{6}&\supset \mathrm {O} (10)\\\mathrm {E} _{7}&\supset \mathrm {O} (12)\\\mathrm {E} _{8}&\supset \mathrm {O} (16)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {U} (n)&\supset \mathrm {O} (n)\\\mathrm {USp} (2n)&\supset \mathrm {O} (n)\\\mathrm {G} _{2}&\supset \mathrm {O} (3)\\\mathrm {F} _{4}&\supset \mathrm {O} (9)\\\mathrm {E} _{6}&\supset \mathrm {O} (10)\\\mathrm {E} _{7}&\supset \mathrm {O} (12)\\\mathrm {E} _{8}&\supset \mathrm {O} (16)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55597f8830d4612981368865f73bca5c348e0ff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:18.487ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {U} (n)&\supset \mathrm {O} (n)\\\mathrm {USp} (2n)&\supset \mathrm {O} (n)\\\mathrm {G} _{2}&\supset \mathrm {O} (3)\\\mathrm {F} _{4}&\supset \mathrm {O} (9)\\\mathrm {E} _{6}&\supset \mathrm {O} (10)\\\mathrm {E} _{7}&\supset \mathrm {O} (12)\\\mathrm {E} _{8}&\supset \mathrm {O} (16)\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Conformal_group">Conformal group</h4>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=34" title="Edit section: Conformal group">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></div> Being <a href="/wiki/Isometry" title="Isometry">isometries</a>, real orthogonal transforms preserve <a href="/wiki/Angle" title="Angle">angles</a>, and are thus <a href="/wiki/Conformal_map" title="Conformal map">conformal maps</a>, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a> and <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a>, as exemplified by SSS (side-side-side) <a href="/wiki/Congruence_(geometry)#Congruence_of_triangles" title="Congruence (geometry)">congruence of triangles</a> and AAA (angle-angle-angle) <a href="/wiki/Similar_triangles" class="mw-redirect" title="Similar triangles">similarity of triangles</a>. The group of conformal linear maps of Rn is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of <a href="/wiki/Homothetic_transformation" class="mw-redirect" title="Homothetic transformation">dilations</a>. If n is odd, these two subgroups do not intersect, and they are a <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a>: CO(2k + 1) = O(2k + 1) × R∗, where R∗ = R∖{0} is the real <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a>, while if n is even, these subgroups intersect in ±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: CO(2k) = O(2k) × R+. Similarly one can define CSO(n); this is always: CSO(n) = CO(n) ∩ GL+(n) = SO(n) × R+. <div class="mw-heading mw-heading3"><h3 id="Discrete_subgroups">Discrete subgroups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=35" title="Edit section: Discrete subgroups">edit</a>]</div> As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.<a href="#cite_note-12">[note 1]</a> These subgroups are known as <a href="/wiki/Point_group" title="Point group">point groups</a> and can be realized as the symmetry groups of <a href="/wiki/Polytope" title="Polytope">polytopes</a>. A very important class of examples are the <a href="/wiki/Finite_Coxeter_group" class="mw-redirect" title="Finite Coxeter group">finite Coxeter groups</a>, which include the symmetry groups of <a href="/wiki/Regular_polytope" title="Regular polytope">regular polytopes</a>. Dimension 3 is particularly studied – see <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a>, <a href="/wiki/Polyhedral_group" title="Polyhedral group">polyhedral groups</a>, and <a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">list of spherical symmetry groups</a>. In 2 dimensions, the finite groups are either cyclic or dihedral – see <a href="/wiki/Point_groups_in_two_dimensions" title="Point groups in two dimensions">point groups in two dimensions</a>. Other finite subgroups include: <ul><li><a href="/wiki/Permutation_matrices" class="mw-redirect" title="Permutation matrices">Permutation matrices</a> (the <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a> An)</li> <li><a href="/wiki/Signed_permutation_matrices" class="mw-redirect" title="Signed permutation matrices">Signed permutation matrices</a> (the <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a> Bn); also equals the intersection of the orthogonal group with the <a href="/wiki/Integer_matrix" title="Integer matrix">integer matrices</a>.<a href="#cite_note-13">[note 2]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Covering_and_quotient_groups">Covering and quotient groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=36" title="Edit section: Covering and quotient groups">edit</a>]</div> The orthogonal group is neither <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> nor <a href="/wiki/Centerless" class="mw-redirect" title="Centerless">centerless</a>, and thus has both a <a href="/wiki/Covering_group" title="Covering group">covering group</a> and a <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a>, respectively: <ul><li>Two covering <a href="/wiki/Pin_group" title="Pin group">Pin groups</a>, Pin+(n) → O(n) and Pin−(n) → O(n),</li> <li>The quotient <a href="/wiki/Projective_orthogonal_group" title="Projective orthogonal group">projective orthogonal group</a>, O(n) → PO(n).</li></ul> These are all 2-to-1 covers. For the special orthogonal group, the corresponding groups are: <ul><li><a href="/wiki/Spin_group" title="Spin group">Spin group</a>, Spin(n) → SO(n),</li> <li><a href="/wiki/Projective_special_orthogonal_group" class="mw-redirect" title="Projective special orthogonal group">Projective special orthogonal group</a>, SO(n) → PSO(n).</li></ul> Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k + 1) is a 1-to-1 cover; i.e., isomorphic to SO(2k + 1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact <a href="/wiki/Special_orthogonal_Lie_algebra" class="mw-redirect" title="Special orthogonal Lie algebra">special orthogonal Lie algebra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(n,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n,\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248649418dbaf041ffc71db6020d773a2cbdd13c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:8.449ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(n,\mathbf {R} )}"> – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.<a href="#cite_note-14">[note 3]</a> In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details. <div class="mw-heading mw-heading2"><h2 id="Principal_homogeneous_space:_Stiefel_manifold">Principal homogeneous space: Stiefel manifold</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=37" title="Edit section: Principal homogeneous space: Stiefel manifold">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stiefel_manifold" title="Stiefel manifold">Stiefel manifold</a></div> The <a href="/wiki/Principal_homogeneous_space" title="Principal homogeneous space">principal homogeneous space</a> for the orthogonal group O(n) is the <a href="/wiki/Stiefel_manifold" title="Stiefel manifold">Stiefel manifold</a> Vn(Rn) of <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal bases</a> (orthonormal <a href="/wiki/K-frame" title="K-frame">n-frames</a>). In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis. The other Stiefel manifolds Vk(Rn) for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=38" title="Edit section: See also">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Specific_transforms">Specific transforms</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=39" title="Edit section: Specific transforms">edit</a>]</div> <ul><li><a href="/wiki/Coordinate_rotations_and_reflections" class="mw-redirect" title="Coordinate rotations and reflections">Coordinate rotations and reflections</a></li> <li><a href="/wiki/Reflection_through_the_origin" class="mw-redirect" title="Reflection through the origin">Reflection through the origin</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Specific_groups">Specific groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=40" title="Edit section: Specific groups">edit</a>]</div> <ul><li>rotation group, <a href="/wiki/Rotation_group_SO(3)" class="mw-redirect" title="Rotation group SO(3)">SO(3, R)</a></li> <li><a href="/wiki/SO(8)" title="SO(8)">SO(8)</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Related_groups_2">Related groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=41" title="Edit section: Related groups">edit</a>]</div> <ul><li><a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">indefinite orthogonal group</a></li> <li><a href="/wiki/Unitary_group" title="Unitary group">unitary group</a></li> <li><a href="/wiki/Symplectic_group" title="Symplectic group">symplectic group</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Lists_of_groups">Lists of groups</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=42" title="Edit section: Lists of groups">edit</a>]</div> <ul><li><a href="/wiki/List_of_finite_simple_groups" title="List of finite simple groups">list of finite simple groups</a></li> <li><a href="/wiki/List_of_simple_Lie_groups" class="mw-redirect" title="List of simple Lie groups">list of simple Lie groups</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Representation_theory">Representation theory</h3>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=43" title="Edit section: Representation theory">edit</a>]</div> <ul><li><a href="/wiki/Representations_of_classical_Lie_groups" title="Representations of classical Lie groups">Representations of classical Lie groups</a></li> <li><a href="/wiki/Brauer_algebra" title="Brauer algebra">Brauer algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=44" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Infinite subsets of a compact space have an <a href="/wiki/Accumulation_point" title="Accumulation point">accumulation point</a> and are not discrete. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> O(n) ∩ <a href="/wiki/General_linear_group" title="General linear group">GL</a>(n, Z) equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> In odd dimension, SO(2k + 1) ≅ PSO(2k + 1) is centerless (but not simply connected), while in even dimension SO(2k) is neither centerless nor simply connected. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=45" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> For base fields of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> not 2, the definition in terms of a <a href="/wiki/Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a> is equivalent to that in terms of a <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a>, but in characteristic 2 these notions differ. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Theorem 11.2 </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 1.3.4 </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Proposition 13.10 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBaez" class="citation web cs1"><a href="/wiki/John_C._Baez" title="John C. Baez">Baez, John</a>. <a rel="nofollow" class="external text" href="https://math.ucr.edu/home/baez/week105.html">"Week 105"</a>. This Week's Finds in Mathematical Physics. Retrieved 2023-02-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=This+Week%27s+Finds+in+Mathematical+Physics&rft.atitle=Week+105&rft.aulast=Baez&rft.aufirst=John&rft_id=https%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fweek105.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"> </li> <li id="cite_note-Wil6975-6">^ <a href="#cite_ref-Wil6975_6-0">a</a> <a href="#cite_ref-Wil6975_6-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson2009" class="citation book cs1">Wilson, Robert A. (2009). The finite simple groups. Graduate Texts in Mathematics. Vol. 251. London: Springer. pp. 69–75. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84800-987-5" title="Special:BookSources/978-1-84800-987-5"><bdi>978-1-84800-987-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1203.20012">1203.20012</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+finite+simple+groups&rft.place=London&rft.series=Graduate+Texts+in+Mathematics&rft.pages=69-75&rft.pub=Springer&rft.date=2009&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1203.20012%23id-name%3DZbl&rft.isbn=978-1-84800-987-5&rft.aulast=Wilson&rft.aufirst=Robert+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> (<a href="#CITEREFTaylor1992">Taylor 1992</a>, p. 141) </li> <li id="cite_note-Knus224-8">^ <a href="#cite_ref-Knus224_8-0">a</a> <a href="#cite_ref-Knus224_8-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnus1991" class="citation cs2">Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, vol. 294, Berlin etc.: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, p. 224, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-52117-8" title="Special:BookSources/3-540-52117-8"><bdi>3-540-52117-8</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0756.11008">0756.11008</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quadratic+and+Hermitian+forms+over+rings&rft.place=Berlin+etc.&rft.series=Grundlehren+der+Mathematischen+Wissenschaften&rft.pages=224&rft.pub=Springer-Verlag&rft.date=1991&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0756.11008%23id-name%3DZbl&rft.isbn=3-540-52117-8&rft.aulast=Knus&rft.aufirst=Max-Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> (<a href="#CITEREFTaylor1992">Taylor 1992</a>, page 160) </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> (<a href="#CITEREFGrove2002">Grove 2002</a>, Theorem 6.6 and 14.16) </li> <li id="cite_note-C178-11"><a href="#cite_ref-C178_11-0">^</a> <a href="#CITEREFCassels1978">Cassels 1978</a>, p. 178 </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=46" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCassels1978" class="citation cs2"><a href="/wiki/J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J.W.S.</a> (1978), Rational Quadratic Forms, London Mathematical Society Monographs, vol. 13, <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-163260-1" title="Special:BookSources/0-12-163260-1"><bdi>0-12-163260-1</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0395.10029">0395.10029</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rational+Quadratic+Forms&rft.series=London+Mathematical+Society+Monographs&rft.pub=Academic+Press&rft.date=1978&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0395.10029%23id-name%3DZbl&rft.isbn=0-12-163260-1&rft.aulast=Cassels&rft.aufirst=J.W.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrove2002" class="citation cs2">Grove, Larry C. (2002), Classical groups and geometric algebra, <a href="/wiki/Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 39, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2019-3" title="Special:BookSources/978-0-8218-2019-3"><bdi>978-0-8218-2019-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1859189">1859189</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+groups+and+geometric+algebra&rft.place=Providence%2C+R.I.&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2002&rft.isbn=978-0-8218-2019-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1859189%23id-name%3DMR&rft.aulast=Grove&rft.aufirst=Larry+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319134666" title="Special:BookSources/978-3319134666"><bdi>978-3319134666</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%2C+Lie+Algebras%2C+and+Representations%3A+An+Elementary+Introduction&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3319134666&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1992" class="citation cs2">Taylor, Donald E. (1992), The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, vol. 9, Berlin: Heldermann Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-88538-009-9" title="Special:BookSources/3-88538-009-9"><bdi>3-88538-009-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1189139">1189139</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0767.20001">0767.20001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+the+Classical+Groups&rft.place=Berlin&rft.series=Sigma+Series+in+Pure+Mathematics&rft.pub=Heldermann+Verlag&rft.date=1992&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0767.20001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1189139%23id-name%3DMR&rft.isbn=3-88538-009-9&rft.aulast=Taylor&rft.aufirst=Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Orthogonal_group&action=edit&section=47" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Orthogonal_group">"Orthogonal group"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Orthogonal+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DOrthogonal_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+group" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week105.html">John Baez "This Week's Finds in Mathematical Physics" week 105</a></li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/octonions/node10.html">John Baez on Octonions</a></li> <li>(in Italian) <a rel="nofollow" class="external text" href="http://ansi.altervista.org">n-dimensional Special Orthogonal Group parametrization</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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Orthogonal group - Wikipedia