Dihedral group - Wikipedia

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class="vector-toc-list"> <li id="toc-Elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Elements</span> </div> </a> <ul id="toc-Elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Group structure</span> </div> </a> <ul id="toc-Group_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Matrix representation</span> </div> </a> <ul id="toc-Matrix_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Other definitions</span> </div> </a> <ul id="toc-Other_definitions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Small_dihedral_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Small_dihedral_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Small dihedral groups</span> </div> </a> <ul id="toc-Small_dihedral_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_dihedral_group_as_symmetry_group_in_2D_and_rotation_group_in_3D" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_dihedral_group_as_symmetry_group_in_2D_and_rotation_group_in_3D"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>The dihedral group as symmetry group in 2D and rotation group in 3D</span> </div> </a> <button aria-controls="toc-The_dihedral_group_as_symmetry_group_in_2D_and_rotation_group_in_3D-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle The dihedral group as symmetry group in 2D and rotation group in 3D subsection</span> </button> <ul id="toc-The_dihedral_group_as_symmetry_group_in_2D_and_rotation_group_in_3D-sublist" class="vector-toc-list"> <li id="toc-Examples_of_2D_dihedral_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_of_2D_dihedral_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Examples of 2D dihedral symmetry</span> </div> </a> <ul id="toc-Examples_of_2D_dihedral_symmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Conjugacy_classes_of_reflections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conjugacy_classes_of_reflections"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Conjugacy classes of reflections</span> </div> </a> <ul id="toc-Conjugacy_classes_of_reflections-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Automorphism_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Automorphism_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Automorphism group</span> </div> </a> <button aria-controls="toc-Automorphism_group-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Automorphism group subsection</span> </button> <ul id="toc-Automorphism_group-sublist" class="vector-toc-list"> <li id="toc-Examples_of_automorphism_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_of_automorphism_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Examples of automorphism groups</span> </div> </a> <ul id="toc-Examples_of_automorphism_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inner_automorphism_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inner_automorphism_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Inner automorphism group</span> </div> </a> <ul id="toc-Inner_automorphism_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </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 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href="https://ca.wikipedia.org/wiki/Grup_diedral" title="Grup diedral – Catalan" lang="ca" hreflang="ca" data-title="Grup diedral" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dihedr%C3%A1ln%C3%AD_grupa" title="Dihedrální grupa – Czech" lang="cs" hreflang="cs" data-title="Dihedrální grupa" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diedergruppe" title="Diedergruppe – German" lang="de" hreflang="de" data-title="Diedergruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_di%C3%A9drico" title="Grupo diédrico – Spanish" lang="es" hreflang="es" data-title="Grupo diédrico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_%D8%AF%D9%88%D9%88%D8%AC%D9%87%DB%8C" title="گروه دووجهی – Persian" lang="fa" hreflang="fa" data-title="گروه دووجهی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_di%C3%A9dral" title="Groupe diédral – French" lang="fr" hreflang="fr" data-title="Groupe diédral" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%9D%B4%EB%A9%B4%EC%B2%B4%EA%B5%B0" title="정이면체군 – Korean" lang="ko" hreflang="ko" data-title="정이면체군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_dihedral" title="Grup dihedral – Indonesian" lang="id" hreflang="id" data-title="Grup dihedral" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Samhverfisgr%C3%BApa_marghyrninga" title="Samhverfisgrúpa marghyrninga – Icelandic" lang="is" hreflang="is" data-title="Samhverfisgrúpa marghyrninga" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_diedrale" title="Gruppo diedrale – Italian" lang="it" hreflang="it" data-title="Gruppo diedrale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%93%D7%99%D7%94%D7%93%D7%A8%D7%9C%D7%99%D7%AA" title="חבורה דיהדרלית – Hebrew" lang="he" hreflang="he" data-title="חבורה דיהדרלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Di%C3%A9dercsoport" title="Diédercsoport – Hungarian" lang="hu" hreflang="hu" data-title="Diédercsoport" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A1%E0%B5%88%E0%B4%B9%E0%B5%86%E0%B4%A1%E0%B5%8D%E0%B4%B0%E0%B5%BD_%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D" title="ഡൈഹെഡ്രൽ ഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ഡൈഹെഡ്രൽ ഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Dihedrale_groep" title="Dihedrale groep – Dutch" lang="nl" hreflang="nl" data-title="Dihedrale groep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%9D%A2%E4%BD%93%E7%BE%A4" title="二面体群 – Japanese" lang="ja" hreflang="ja" data-title="二面体群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_diedralna" title="Grupa diedralna – Polish" lang="pl" hreflang="pl" data-title="Grupa diedralna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_diedral" title="Grupo diedral – Portuguese" lang="pt" hreflang="pt" data-title="Grupo diedral" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Grup_diedral" title="Grup diedral – Romanian" lang="ro" hreflang="ro" data-title="Grup diedral" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%8D%D0%B4%D1%80%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="Диэдральная группа – Russian" lang="ru" hreflang="ru" data-title="Диэдральная группа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Diedrska_grupa" title="Diedrska grupa – Slovenian" lang="sl" hreflang="sl" data-title="Diedrska grupa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diedriryhm%C3%A4" title="Diedriryhmä – Finnish" lang="fi" hreflang="fi" data-title="Diedriryhmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Dihedral_grupp" title="Dihedral grupp – Swedish" lang="sv" hreflang="sv" data-title="Dihedral grupp" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AF%8D" title="இருமுகக் குலங்கள் – Tamil" lang="ta" hreflang="ta" data-title="இருமுகக் குலங்கள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%B5%D0%B4%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Діедрична група – Ukrainian" lang="uk" hreflang="uk" data-title="Діедрична група" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_nh%E1%BB%8B_di%E1%BB%87n" title="Nhóm nhị diện – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm nhị diện" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh 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"><span typeof="mw:File"><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></span></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"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></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 class="mw-selflink selflink">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"><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> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a class="mw-selflink selflink">Dihedral group</a> D<sub><i>n</i></sub></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"><i>p</i>-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> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</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, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</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 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/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(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Snowflake8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Snowflake8.png/220px-Snowflake8.png" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Snowflake8.png/330px-Snowflake8.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3f/Snowflake8.png 2x" data-file-width="341" data-file-height="319" /></a><figcaption>The <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of a <a href="/wiki/Snowflake" title="Snowflake">snowflake</a> is D<sub>6</sub>, a dihedral symmetry, the same as for a regular <a href="/wiki/Hexagon" title="Hexagon">hexagon</a>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>dihedral group</b> is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Symmetry" title="Symmetry">symmetries</a> of a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which includes <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotations</a> and <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflections</a>. Dihedral groups are among the simplest examples of <a href="/wiki/Finite_group" title="Finite group">finite groups</a>, and they play an important role in <a href="/wiki/Group_theory" title="Group theory">group theory</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>. </p><p>The notation for the dihedral group differs in <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. In <a href="/wiki/Geometry" title="Geometry">geometry</a>, <span class="texhtml">D<sub><i>n</i></sub></span> or <span class="texhtml">Dih<sub><i>n</i></sub></span> refers to the symmetries of the <a href="/wiki/N-gon" class="mw-redirect" title="N-gon"><span class="texhtml"><i>n</i></span>-gon</a>, a group of order <span class="texhtml">2<i>n</i></span>. In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, <span class="texhtml">D<sub>2<i>n</i></sub></span> refers to this same dihedral group.<sup id="cite_ref-mathimages_3-0" class="reference"><a href="#cite_note-mathimages-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This article uses the geometric convention, <span class="texhtml">D<sub><i>n</i></sub></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word "dihedral" comes from "di-" and "-hedron". The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall it thus refers to the two faces of a polygon. </p> <div class="mw-heading mw-heading3"><h3 id="Elements">Elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=2" title="Edit section: Elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hexagon_reflections.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hexagon_reflections.svg/220px-Hexagon_reflections.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hexagon_reflections.svg/330px-Hexagon_reflections.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hexagon_reflections.svg/440px-Hexagon_reflections.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>The six axes of <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection</a> of a regular hexagon</figcaption></figure> <p>A regular polygon with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> sides has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}"></span> different symmetries: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetries</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetries</a>. Usually, we take <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73136e4a27fe39c123d16a7808e76d3162ce42bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}"></span> here. The associated <a href="/wiki/Rotation" title="Rotation">rotations</a> and <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a> make up the dihedral group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded8c7d71e610ba30a0856fa881290ae80b7282b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.994ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{n}}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is even, there are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c3931a3fa03cc98cfacd2c49a7ca35b96eaa9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle n/2}"></span> axes of symmetry connecting the midpoints of opposite sides and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c3931a3fa03cc98cfacd2c49a7ca35b96eaa9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle n/2}"></span> axes of symmetry connecting opposite vertices. In either case, there are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> axes of symmetry and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}"></span> elements in the symmetry group.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Dihedral8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Dihedral8.png/550px-Dihedral8.png" decoding="async" width="550" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/96/Dihedral8.png 1.5x" data-file-width="710" data-file-height="196" /></a><figcaption>The following picture shows the effect of the sixteen elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3d021f35eeff0b18797d343c193a289271b7c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{8}}"></span> on a <a href="/wiki/Stop_sign" title="Stop sign">stop sign</a>. Here, the first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Group_structure">Group structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=3" title="Edit section: Group structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As with any geometric object, the <a href="/wiki/Composition_of_functions" class="mw-redirect" title="Composition of functions">composition</a> of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a <a href="/wiki/Finite_group" title="Finite group">finite group</a>.<sup id="cite_ref-lovett_6-0" class="reference"><a href="#cite_note-lovett-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Labeled_Triangle_Reflections.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Labeled_Triangle_Reflections.svg/220px-Labeled_Triangle_Reflections.svg.png" decoding="async" width="220" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Labeled_Triangle_Reflections.svg/330px-Labeled_Triangle_Reflections.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Labeled_Triangle_Reflections.svg/440px-Labeled_Triangle_Reflections.svg.png 2x" data-file-width="870" data-file-height="754" /></a><figcaption>The lines of reflection labelled S<sub>0</sub>, S<sub>1</sub>, and S<sub>2</sub> remain fixed in space (on the page) and do not themselves move when a symmetry operation (rotation or reflection) is done on the triangle (this matters when doing compositions of symmetries).</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Two_Reflection_Rotation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Two_Reflection_Rotation.svg/220px-Two_Reflection_Rotation.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Two_Reflection_Rotation.svg/330px-Two_Reflection_Rotation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Two_Reflection_Rotation.svg/440px-Two_Reflection_Rotation.svg.png 2x" data-file-width="870" data-file-height="629" /></a><figcaption>The composition of these two reflections is a rotation.</figcaption></figure> <p>The following <a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a> shows the effect of composition in the group <a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">D<sub>3</sub></a> (the symmetries of an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>). r<sub>0</sub> denotes the identity; r<sub>1</sub> and r<sub>2</sub> denote counterclockwise rotations by 120° and 240° respectively, and s<sub>0</sub>, s<sub>1</sub> and s<sub>2</sub> denote reflections across the three lines shown in the adjacent picture. </p> <table class="wikitable" width="200"> <tbody><tr> <th></th> <th>r<sub>0</sub></th> <th>r<sub>1</sub></th> <th>r<sub>2</sub></th> <th>s<sub>0</sub></th> <th>s<sub>1</sub></th> <th>s<sub>2</sub> </th></tr> <tr> <th>r<sub>0</sub> </th> <td>r<sub>0</sub></td> <td>r<sub>1</sub></td> <td>r<sub>2</sub> </td> <td>s<sub>0</sub></td> <td>s<sub>1</sub></td> <td>s<sub>2</sub> </td></tr> <tr> <th>r<sub>1</sub> </th> <td>r<sub>1</sub></td> <td>r<sub>2</sub></td> <td>r<sub>0</sub> </td> <td>s<sub>1</sub></td> <td>s<sub>2</sub></td> <td>s<sub>0</sub> </td></tr> <tr> <th>r<sub>2</sub> </th> <td>r<sub>2</sub></td> <td>r<sub>0</sub></td> <td>r<sub>1</sub> </td> <td>s<sub>2</sub></td> <td>s<sub>0</sub></td> <td>s<sub>1</sub> </td></tr> <tr> <th>s<sub>0</sub> </th> <td>s<sub>0</sub></td> <td>s<sub>2</sub></td> <td>s<sub>1</sub> </td> <td>r<sub>0</sub></td> <td>r<sub>2</sub></td> <td>r<sub>1</sub> </td></tr> <tr> <th>s<sub>1</sub> </th> <td>s<sub>1</sub></td> <td>s<sub>0</sub></td> <td>s<sub>2</sub> </td> <td>r<sub>1</sub></td> <td>r<sub>0</sub></td> <td>r<sub>2</sub> </td></tr> <tr> <th>s<sub>2</sub> </th> <td>s<sub>2</sub></td> <td>s<sub>1</sub></td> <td>s<sub>0</sub> </td> <td>r<sub>2</sub></td> <td>r<sub>1</sub></td> <td>r<sub>0</sub> </td></tr></tbody></table> <p>For example, <span class="nowrap">s<sub>2</sub>s<sub>1</sub> = r<sub>1</sub></span>, because the reflection s<sub>1</sub> followed by the reflection s<sub>2</sub> results in a rotation of 120°. The order of elements denoting the <a href="/wiki/Composition_of_functions" class="mw-redirect" title="Composition of functions">composition</a> is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutative</a>.<sup id="cite_ref-lovett_6-1" class="reference"><a href="#cite_note-lovett-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In general, the group D<sub><i>n</i></sub> has elements r<sub>0</sub>, ..., r<sub><i>n</i>−1</sub> and s<sub>0</sub>, ..., s<sub><i>n</i>−1</sub>, with composition given by the following formulae: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},\quad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\quad \mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},\quad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},\quad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\quad \mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},\quad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466a23846d33c9b68491e87595406476d76fa9a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.486ex; height:2.343ex;" alt="{\displaystyle \mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},\quad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\quad \mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},\quad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.}"></span></dd></dl> <p>In all cases, addition and subtraction of subscripts are to be performed using <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> with modulus <i>n</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_representation">Matrix representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=4" title="Edit section: Matrix representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pentagon_Linear.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Pentagon_Linear.png/220px-Pentagon_Linear.png" decoding="async" width="220" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/59/Pentagon_Linear.png 1.5x" data-file-width="319" data-file-height="311" /></a><figcaption>The symmetries of this pentagon are <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> of the plane as a vector space.</figcaption></figure> <p>If we center the regular polygon at the origin, then elements of the dihedral group act as <a href="/wiki/Linear_map" title="Linear map">linear transformations</a> of the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">plane</a>. This lets us represent elements of D<sub><i>n</i></sub> as <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, with composition being <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. This is an example of a (2-dimensional) <a href="/wiki/Group_representation" title="Group representation">group representation</a>. </p><p>For example, the elements of the group <a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">D<sub>4</sub></a> can be represented by the following eight matrices: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\mathrm {r} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {r} _{1}=\left({\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {r} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {r} _{3}=\left({\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}}\right),\\[1em]\mathrm {s} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {s} _{1}=\left({\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {s} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {s} _{3}=\left({\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}}\right).\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="1.4em 0.4em" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <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> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <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> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing="0.4em 0.2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\mathrm {r} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {r} _{1}=\left({\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {r} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {r} _{3}=\left({\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}}\right),\\[1em]\mathrm {s} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {s} _{1}=\left({\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {s} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {s} _{3}=\left({\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}}\right).\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72296b4c5f5de93a62ee7d535c60589b3da46cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:63.57ex; height:12.176ex;" alt="{\displaystyle {\begin{matrix}\mathrm {r} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {r} _{1}=\left({\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {r} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {r} _{3}=\left({\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}}\right),\\[1em]\mathrm {s} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {s} _{1}=\left({\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {s} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {s} _{3}=\left({\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}}\right).\end{matrix}}}"></span></dd></dl> <p>In general, the matrices for elements of D<sub><i>n</i></sub> have the following form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {r} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&-\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&\cos {\frac {2\pi k}{n}}\end{pmatrix}}\ \ {\text{and}}\\[5pt]\mathrm {s} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&-\cos {\frac {2\pi k}{n}}\end{pmatrix}}.\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="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {r} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&-\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&\cos {\frac {2\pi k}{n}}\end{pmatrix}}\ \ {\text{and}}\\[5pt]\mathrm {s} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&-\cos {\frac {2\pi k}{n}}\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df538c73fe0c524d36aebb6da4a6b9df94c97db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:33.368ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {r} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&-\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&\cos {\frac {2\pi k}{n}}\end{pmatrix}}\ \ {\text{and}}\\[5pt]\mathrm {s} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&-\cos {\frac {2\pi k}{n}}\end{pmatrix}}.\end{aligned}}}"></span></dd></dl> <p>r<sub><i>k</i></sub> is a <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a>, expressing a counterclockwise rotation through an angle of <span class="nowrap">2<i>πk</i>/<i>n</i></span>. s<sub><i>k</i></sub> is a reflection across a line that makes an angle of <span class="nowrap"><i>πk</i>/<i>n</i></span> with the <i>x</i>-axis. </p> <div class="mw-heading mw-heading3"><h3 id="Other_definitions">Other definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=5" title="Edit section: Other definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="texhtml">D<sub><i>n</i></sub></span> can also be defined as the group with <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>∣</mo> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>,</mo> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>s</mi> <mi>r</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>∣</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e37578e4af143c4d3a7282442dea30271f304db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:49.504ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\end{aligned}}}"></span></dd></dl> <p>Using the relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23d62feea32b1ca59d539beb588e05c83b499aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.406ex; height:2.676ex;" alt="{\displaystyle s^{2}=1}"></span>, we obtain the relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=s\cdot sr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mi>s</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=s\cdot sr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5ba07469fbbed09e79360216360163e69ccf5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.056ex; height:1.676ex;" alt="{\displaystyle r=s\cdot sr}"></span>. It follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded8c7d71e610ba30a0856fa881290ae80b7282b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.994ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{n}}"></span> is generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t:=sr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>:=</mo> <mi>s</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t:=sr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63dd9de18b07962cab2469c0395bad0f50b9caff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.724ex; height:2.009ex;" alt="{\displaystyle t:=sr}"></span>. This substitution also shows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded8c7d71e610ba30a0856fa881290ae80b7282b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.994ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{n}}"></span> has the presentation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{n}=\left\langle s,t\mid s^{2}=1,t^{2}=1,(st)^{n}=1\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∣</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{n}=\left\langle s,t\mid s^{2}=1,t^{2}=1,(st)^{n}=1\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0fdaf90d87c20ff1d84b9a30a2f9bee2b591ca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.07ex; height:3.343ex;" alt="{\displaystyle \mathrm {D} _{n}=\left\langle s,t\mid s^{2}=1,t^{2}=1,(st)^{n}=1\right\rangle .}"></span></dd></dl> <p>In particular, <span class="texhtml">D<sub><i>n</i></sub></span> belongs to the class of <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Small_dihedral_groups">Small dihedral groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=6" title="Edit section: Small dihedral groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Regular_hexagon_symmetries.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Regular_hexagon_symmetries.svg/350px-Regular_hexagon_symmetries.svg.png" decoding="async" width="350" height="303" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Regular_hexagon_symmetries.svg/525px-Regular_hexagon_symmetries.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Regular_hexagon_symmetries.svg/700px-Regular_hexagon_symmetries.svg.png 2x" data-file-width="886" data-file-height="766" /></a><figcaption>Example subgroups from a hexagonal dihedral symmetry</figcaption></figure> <p><span class="texhtml">D<sub>1</sub></span> is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to <span class="texhtml">Z<sub>2</sub></span>, the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of order 2. </p><p><span class="texhtml">D<sub>2</sub></span> is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to <span class="texhtml">K<sub>4</sub></span>, the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a>. </p><p><span class="texhtml">D<sub>1</sub></span> and <span class="texhtml">D<sub>2</sub></span> are exceptional in that: </p> <ul><li><span class="texhtml">D<sub>1</sub></span> and <span class="texhtml">D<sub>2</sub></span> are the only <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> dihedral groups. Otherwise, <span class="texhtml">D<sub><i>n</i></sub></span> is non-abelian.</li> <li><span class="texhtml">D<sub><i>n</i></sub></span> is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> <span class="texhtml">S<sub><i>n</i></sub></span> for <span class="texhtml"><i>n</i> ≥ 3</span>. Since <span class="texhtml">2<i>n</i> > <i>n</i>!</span> for <span class="texhtml"><i>n</i> = 1</span> or <span class="texhtml"><i>n</i> = 2</span>, for these values, <span class="texhtml">D<sub><i>n</i></sub></span> is too large to be a subgroup.</li> <li>The inner automorphism group of <span class="texhtml">D<sub>2</sub></span> is trivial, whereas for other even values of <span class="texhtml"><i>n</i></span>, this is <span class="texhtml">D<sub><i>n</i></sub> / Z<sub>2</sub></span>.</li></ul> <p>The <a href="/wiki/Cycle_graph_(group)" class="mw-redirect" title="Cycle graph (group)">cycle graphs</a> of dihedral groups consist of an <i>n</i>-element cycle and <i>n</i> 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the <a href="/wiki/Identity_element" title="Identity element">identity element</a>. </p> <table class="wikitable"> <caption>Cycle graphs </caption> <tbody><tr> <th>D<sub>1</sub> = <a href="/wiki/Cyclic_group" title="Cyclic group">Z<sub>2</sub></a></th> <th>D<sub>2</sub> = Z<sub>2</sub><sup>2</sup> = <a href="/wiki/Klein_four-group" title="Klein four-group">K<sub>4</sub></a></th> <th>D<sub>3</sub></th> <th>D<sub>4</sub></th> <th>D<sub>5</sub> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/100px-GroupDiagramMiniC2.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/150px-GroupDiagramMiniC2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/200px-GroupDiagramMiniC2.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/100px-GroupDiagramMiniD4.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/150px-GroupDiagramMiniD4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/200px-GroupDiagramMiniD4.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/100px-GroupDiagramMiniD6.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/150px-GroupDiagramMiniD6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/200px-GroupDiagramMiniD6.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/100px-GroupDiagramMiniD8.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/150px-GroupDiagramMiniD8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/200px-GroupDiagramMiniD8.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/100px-GroupDiagramMiniD10.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/150px-GroupDiagramMiniD10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/200px-GroupDiagramMiniD10.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/100px-GroupDiagramMiniD12.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/150px-GroupDiagramMiniD12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/200px-GroupDiagramMiniD12.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD14.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/100px-GroupDiagramMiniD14.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/150px-GroupDiagramMiniD14.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/200px-GroupDiagramMiniD14.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/100px-GroupDiagramMiniD16.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/150px-GroupDiagramMiniD16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/200px-GroupDiagramMiniD16.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD18.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/GroupDiagramMiniD18.png/100px-GroupDiagramMiniD18.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/GroupDiagramMiniD18.png/150px-GroupDiagramMiniD18.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/GroupDiagramMiniD18.png/200px-GroupDiagramMiniD18.png 2x" data-file-width="762" data-file-height="735" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD20.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/GroupDiagramMiniD20.png/100px-GroupDiagramMiniD20.png" decoding="async" width="100" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/GroupDiagramMiniD20.png/150px-GroupDiagramMiniD20.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/GroupDiagramMiniD20.png/200px-GroupDiagramMiniD20.png 2x" data-file-width="749" data-file-height="817" /></a></span> </td></tr> <tr> <th>D<sub>6</sub> = D<sub>3</sub> × Z<sub>2</sub></th> <th>D<sub>7</sub></th> <th>D<sub>8</sub></th> <th>D<sub>9</sub></th> <th>D<sub>10</sub> = D<sub>5</sub> × Z<sub>2</sub> </th></tr></tbody></table> <table class="wikitable" style="text-align:center; vertical-align:top;"> <tbody><tr> <th><a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">D<sub>3</sub></a> = <a href="/wiki/Symmetric_group" title="Symmetric group">S</a><sub>3</sub> </th> <th><a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">D<sub>4</sub></a> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Symmetric_group_3;_cycle_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Symmetric_group_3%3B_cycle_graph.svg/240px-Symmetric_group_3%3B_cycle_graph.svg.png" decoding="async" width="240" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Symmetric_group_3%3B_cycle_graph.svg/360px-Symmetric_group_3%3B_cycle_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Symmetric_group_3%3B_cycle_graph.svg/480px-Symmetric_group_3%3B_cycle_graph.svg.png 2x" data-file-width="1662" data-file-height="1910" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Dih4_cycle_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Dih4_cycle_graph.svg/240px-Dih4_cycle_graph.svg.png" decoding="async" width="240" height="307" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Dih4_cycle_graph.svg/360px-Dih4_cycle_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Dih4_cycle_graph.svg/480px-Dih4_cycle_graph.svg.png 2x" data-file-width="1172" data-file-height="1501" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="The_dihedral_group_as_symmetry_group_in_2D_and_rotation_group_in_3D">The dihedral group as symmetry group in 2D and rotation group in 3D</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=7" title="Edit section: The dihedral group as symmetry group in 2D and rotation group in 3D"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An example of abstract group <span class="texhtml">D<sub><i>n</i></sub></span>, and a common way to visualize it, is the group of <a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean plane isometries</a> which keep the origin fixed. These groups form one of the two series of discrete <a href="/wiki/Point_groups_in_two_dimensions" title="Point groups in two dimensions">point groups in two dimensions</a>. <span class="texhtml">D<sub><i>n</i></sub></span> consists of <span class="texhtml"><i>n</i></span> <a href="/wiki/Rotation" title="Rotation">rotations</a> of multiples of <span class="texhtml">360°/<i>n</i></span> about the origin, and <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a> across <span class="texhtml"><i>n</i></span> lines through the origin, making angles of multiples of <span class="texhtml">180°/<i>n</i></span> with each other. This is the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a> with <span class="texhtml"><i>n</i></span> sides (for <span class="texhtml"><i>n</i> ≥ 3</span>; this extends to the cases <span class="texhtml"><i>n</i> = 1</span> and <span class="texhtml"><i>n</i> = 2</span> where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment). </p><p><span class="texhtml">D<sub><i>n</i></sub></span> is <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by a rotation <span class="texhtml">r</span> of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> <span class="texhtml"><i>n</i></span> and a reflection <span class="texhtml">s</span> of order 2 such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {srs} =\mathrm {r} ^{-1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">s</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {srs} =\mathrm {r} ^{-1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea07738a5644b7f1d1eb2226183ff9591b86fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.474ex; height:2.676ex;" alt="{\displaystyle \mathrm {srs} =\mathrm {r} ^{-1}\,}"></span></dd></dl> <p>In geometric terms: in the mirror a rotation looks like an inverse rotation. </p><p>In terms of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>: multiplication by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2\pi i \over n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2\pi i \over n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/babca7ffae1bf427a869d33a130bd911066aa38d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.045ex; height:3.343ex;" alt="{\displaystyle e^{2\pi i \over n}}"></span> and <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. </p><p>In matrix form, by setting </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {r} _{1}={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\[4pt]\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad \mathrm {s} _{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="0.8em 0.4em" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="2em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {r} _{1}={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\[4pt]\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad \mathrm {s} _{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8bff7dfa39121ba3bd82f144fa7b762715e93c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.751ex; height:8.176ex;" alt="{\displaystyle \mathrm {r} _{1}={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\[4pt]\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad \mathrm {s} _{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}"></span></dd></dl> <p>and defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {r} _{j}=\mathrm {r} _{1}^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {r} _{j}=\mathrm {r} _{1}^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7257eabfcf1c12a0ed0062bf1b99f076dfafb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.886ex; height:3.509ex;" alt="{\displaystyle \mathrm {r} _{j}=\mathrm {r} _{1}^{j}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {s} _{j}=\mathrm {r} _{j}\,\mathrm {s} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {s} _{j}=\mathrm {r} _{j}\,\mathrm {s} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ede00c05af70f5f79615da220c3c676325e45f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.103ex; height:2.343ex;" alt="{\displaystyle \mathrm {s} _{j}=\mathrm {r} _{j}\,\mathrm {s} _{0}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in \{1,\ldots ,n-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\in \{1,\ldots ,n-1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640da579f088fe626b00a354c57109ae551cc83d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:17.889ex; height:2.843ex;" alt="{\displaystyle j\in \{1,\ldots ,n-1\}}"></span> we can write the product rules for D<sub><i>n</i></sub> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {r} _{j}\,\mathrm {r} _{k}&=\mathrm {r} _{(j+k){\text{ mod }}n}\\\mathrm {r} _{j}\,\mathrm {s} _{k}&=\mathrm {s} _{(j+k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {r} _{k}&=\mathrm {s} _{(j-k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {s} _{k}&=\mathrm {r} _{(j-k){\text{ mod }}n}\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> mod </mtext> </mrow> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> mod </mtext> </mrow> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> mod </mtext> </mrow> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> mod </mtext> </mrow> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {r} _{j}\,\mathrm {r} _{k}&=\mathrm {r} _{(j+k){\text{ mod }}n}\\\mathrm {r} _{j}\,\mathrm {s} _{k}&=\mathrm {s} _{(j+k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {r} _{k}&=\mathrm {s} _{(j-k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {s} _{k}&=\mathrm {r} _{(j-k){\text{ mod }}n}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf6f9885f4aff5cb52a3f0a10d2a8e030ccaebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:18.561ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\mathrm {r} _{j}\,\mathrm {r} _{k}&=\mathrm {r} _{(j+k){\text{ mod }}n}\\\mathrm {r} _{j}\,\mathrm {s} _{k}&=\mathrm {s} _{(j+k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {r} _{k}&=\mathrm {s} _{(j-k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {s} _{k}&=\mathrm {r} _{(j-k){\text{ mod }}n}\end{aligned}}}"></span></dd></dl> <p>(Compare <a href="/wiki/Coordinate_rotations_and_reflections" class="mw-redirect" title="Coordinate rotations and reflections">coordinate rotations and reflections</a>.) </p><p>The dihedral group D<sub>2</sub> is generated by the rotation r of 180 degrees, and the reflection s across the <i>x</i>-axis. The elements of D<sub>2</sub> can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the <i>y</i>-axis. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dihedral4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b6/Dihedral4.png" decoding="async" width="485" height="124" class="mw-file-element" data-file-width="485" data-file-height="124" /></a><figcaption>The four elements of D<sub>2</sub> (x-axis is vertical here)</figcaption></figure> <p>D<sub>2</sub> is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a>. </p><p>For <i>n</i> > 2 the operations of rotation and reflection in general do not <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commute</a> and D<sub><i>n</i></sub> is not <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>; for example, in <a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">D<sub>4</sub></a>, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:D8isNonAbelian.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/20/D8isNonAbelian.png" decoding="async" width="485" height="124" class="mw-file-element" data-file-width="485" data-file-height="124" /></a><figcaption>D<sub>4</sub> is nonabelian (x-axis is vertical here).</figcaption></figure> <p>Thus, beyond their obvious application to problems of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups. </p><p>The <span class="texhtml">2<i>n</i></span> elements of <span class="texhtml">D<sub><i>n</i></sub></span> can be written as <span class="texhtml">e</span>, <span class="texhtml">r</span>, <span class="texhtml">r<sup>2</sup></span>, ... , <span class="texhtml">r<sup><i>n</i>−1</sup></span>, <span class="texhtml">s</span>, <span class="texhtml">r s</span>, <span class="texhtml">r<sup>2</sup>s</span>, ... , <span class="texhtml">r<sup><i>n</i>−1</sup>s</span>. The first <span class="texhtml"><i>n</i></span> listed elements are rotations and the remaining <span class="texhtml"><i>n</i></span> elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection. </p><p>So far, we have considered <span class="texhtml">D<sub><i>n</i></sub></span> to be a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of <span class="texhtml"><a href="/wiki/Orthogonal_group" title="Orthogonal group">O(2)</a></span>, i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation <span class="texhtml">D<sub><i>n</i></sub></span> is also used for a subgroup of <a href="/wiki/SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a> which is also of abstract group type <span class="texhtml">D<sub><i>n</i></sub></span>: the <a href="/wiki/Symmetry_group" title="Symmetry group">proper symmetry group</a> of a <i>regular polygon embedded in three-dimensional space</i> (if <i>n</i> ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore, it is also called a <i>dihedron</i> (Greek: solid with two faces), which explains the name <i>dihedral group</i> (in analogy to <i>tetrahedral</i>, <i>octahedral</i> and <i>icosahedral group</i>, referring to the proper symmetry groups of a regular <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, and <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> respectively). </p> <div class="mw-heading mw-heading3"><h3 id="Examples_of_2D_dihedral_symmetry">Examples of 2D dihedral symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=8" title="Edit section: Examples of 2D dihedral symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Imperial_Seal_of_Japan.svg" class="mw-file-description" title="2D D16 symmetry – Imperial Seal of Japan, representing eightfold chrysanthemum with sixteen petals."><img alt="2D D16 symmetry – Imperial Seal of Japan, representing eightfold chrysanthemum with sixteen petals." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Imperial_Seal_of_Japan.svg/120px-Imperial_Seal_of_Japan.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Imperial_Seal_of_Japan.svg/180px-Imperial_Seal_of_Japan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Imperial_Seal_of_Japan.svg/240px-Imperial_Seal_of_Japan.svg.png 2x" data-file-width="990" data-file-height="990" /></a></span></div> <div class="gallerytext">2D D<sub>16</sub> symmetry – Imperial Seal of Japan, representing eightfold <a href="/wiki/Chrysanthemum" title="Chrysanthemum">chrysanthemum</a> with sixteen <a href="/wiki/Petal" title="Petal">petals</a>.</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Red_Star_of_David.svg" class="mw-file-description" title="2D D6 symmetry – The Red Star of David"><img alt="2D D6 symmetry – The Red Star of David" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Red_Star_of_David.svg/180px-Red_Star_of_David.svg.png" decoding="async" width="180" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Red_Star_of_David.svg/270px-Red_Star_of_David.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Red_Star_of_David.svg/360px-Red_Star_of_David.svg.png 2x" data-file-width="900" data-file-height="600" /></a></span></div> <div class="gallerytext">2D D<sub>6</sub> symmetry – <a href="/wiki/Magen_David_Adom" title="Magen David Adom">The Red Star of David</a></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Naval_Jack_of_the_Republic_of_China.svg" class="mw-file-description" title="2D D12 symmetry — The Naval Jack of the Republic of China (White Sun)"><img alt="2D D12 symmetry — The Naval Jack of the Republic of China (White Sun)" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Naval_Jack_of_the_Republic_of_China.svg/180px-Naval_Jack_of_the_Republic_of_China.svg.png" decoding="async" width="180" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Naval_Jack_of_the_Republic_of_China.svg/270px-Naval_Jack_of_the_Republic_of_China.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Naval_Jack_of_the_Republic_of_China.svg/360px-Naval_Jack_of_the_Republic_of_China.svg.png 2x" data-file-width="900" data-file-height="600" /></a></span></div> <div class="gallerytext">2D D<sub>12</sub> symmetry — The Naval Jack of the Republic of China (White Sun)</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Ashoka_Chakra.svg" class="mw-file-description" title="2D D24 symmetry – Ashoka Chakra, as depicted on the National flag of the Republic of India."><img alt="2D D24 symmetry – Ashoka Chakra, as depicted on the National flag of the Republic of India." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Ashoka_Chakra.svg/120px-Ashoka_Chakra.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Ashoka_Chakra.svg/180px-Ashoka_Chakra.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Ashoka_Chakra.svg/240px-Ashoka_Chakra.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span></div> <div class="gallerytext">2D D<sub>24</sub> symmetry – <a href="/wiki/Ashoka_Chakra" title="Ashoka Chakra">Ashoka Chakra</a>, as depicted on the <a href="/wiki/Flag_of_India" title="Flag of India">National flag of the Republic of India</a>. </div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=9" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The properties of the dihedral groups <span class="texhtml">D<sub><i>n</i></sub></span> with <span class="texhtml"><i>n</i> ≥ 3</span> depend on whether <span class="texhtml"><i>n</i></span> is even or odd. For example, the <a href="/wiki/Center_of_a_group" class="mw-redirect" title="Center of a group">center</a> of <span class="texhtml">D<sub><i>n</i></sub></span> consists only of the identity if <i>n</i> is odd, but if <i>n</i> is even the center has two elements, namely the identity and the element r<sup><i>n</i>/2</sup> (with D<sub><i>n</i></sub> as a subgroup of O(2), this is <a href="/wiki/Inversion_(discrete_mathematics)" title="Inversion (discrete mathematics)">inversion</a>; since it is <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> by −1, it is clear that it commutes with any linear transformation). </p><p>In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones. </p><p>For <i>n</i> twice an odd number, the abstract group <span class="texhtml">D<sub><i>n</i></sub></span> is isomorphic with the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of <span class="texhtml">D<sub><i>n</i> / 2</sub></span> and <span class="texhtml">Z<sub>2</sub></span>. Generally, if <i>m</i> <a href="/wiki/Divisor" title="Divisor">divides</a> <i>n</i>, then <span class="texhtml">D<sub><i>n</i></sub></span> has <i>n</i>/<i>m</i> <a href="/wiki/Subgroup" title="Subgroup">subgroups</a> of type <span class="texhtml">D<sub><i>m</i></sub></span>, and one subgroup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span><sub><i>m</i></sub>. Therefore, the total number of subgroups of <span class="texhtml">D<sub><i>n</i></sub></span> (<i>n</i> ≥ 1), is equal to <i>d</i>(<i>n</i>) + σ(<i>n</i>), where <i>d</i>(<i>n</i>) is the number of positive <a href="/wiki/Divisor" title="Divisor">divisors</a> of <i>n</i> and <i>σ</i>(<i>n</i>) is the sum of the positive divisors of <i>n</i>. See <a href="/wiki/List_of_small_groups" title="List of small groups">list of small groups</a> for the cases <i>n</i> ≤ 8. </p><p>The dihedral group of order 8 (D<sub>4</sub>) is the smallest example of a group that is not a <a href="/wiki/T-group_(mathematics)" title="T-group (mathematics)">T-group</a>. Any of its two <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a> subgroups (which are normal in D<sub>4</sub>) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D<sub>4</sub>, but these subgroups are not normal in D<sub>4</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Conjugacy_classes_of_reflections">Conjugacy classes of reflections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=10" title="Edit section: Conjugacy classes of reflections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All the reflections are <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugate</a> to each other whenever <i>n</i> is odd, but they fall into two conjugacy classes if <i>n</i> is even. If we think of the isometries of a regular <i>n</i>-gon: for odd <i>n</i> there are rotations in the group between every pair of mirrors, while for even <i>n</i> only half of the mirrors can be reached from one by these rotations. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon there are two sets of axes, each corresponding to a conjugacy class: those that pass through two vertices and those that pass through two sides. </p><p>Algebraically, this is an instance of the conjugate <a href="/wiki/Sylow_theorem" class="mw-redirect" title="Sylow theorem">Sylow theorem</a> (for <i>n</i> odd): for <i>n</i> odd, each reflection, together with the identity, form a subgroup of order 2, which is a <a href="/wiki/Sylow_subgroup" class="mw-redirect" title="Sylow subgroup">Sylow 2-subgroup</a> (<span class="nowrap">2 = 2<sup>1</sup></span> is the maximum power of 2 dividing <span class="nowrap">2<i>n</i> = 2[2<i>k</i> + 1]</span>), while for <i>n</i> even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group. </p><p>For <i>n</i> even there is instead an <a href="/wiki/Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer automorphism</a> interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism). </p> <div class="mw-heading mw-heading2"><h2 id="Automorphism_group">Automorphism group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=11" title="Edit section: Automorphism group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of <span class="texhtml">D<sub><i>n</i></sub></span> is isomorphic to the <a href="/wiki/Holomorph_(mathematics)" title="Holomorph (mathematics)">holomorph</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>/<i>n</i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>, i.e., to <span class="nowrap">Hol(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>/<i>n</i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>) = {<i>ax</i> + <i>b</i> | (<i>a</i>, <i>n</i>) = 1} </span> and has order <i>nϕ</i>(<i>n</i>), where <i>ϕ</i> is Euler's <a href="/wiki/Totient" class="mw-redirect" title="Totient">totient</a> function, the number of <i>k</i> in <span class="nowrap">1, ..., <i>n</i> − 1</span> coprime to <i>n</i>. </p><p>It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by <i>k</i>(2<i>π</i>/<i>n</i>), for <i>k</i> <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to <i>n</i>); which automorphisms are inner and outer depends on the parity of <i>n</i>. </p> <ul><li>For <i>n</i> odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for <i>n</i> even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center.</li> <li>Thus for <i>n</i> odd, the inner automorphism group has order 2<i>n</i>, and for <i>n</i> even (other than <span class="nowrap"><i>n</i> = 2</span>) the inner automorphism group has order <i>n</i>.</li> <li>For <i>n</i> odd, all reflections are conjugate; for <i>n</i> even, they fall into two classes (those through two vertices and those through two faces), related by an outer automorphism, which can be represented by rotation by <i>π</i>/<i>n</i> (half the minimal rotation).</li> <li>The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, but otherwise leaves them unchanged. Thus automorphisms that multiply angles by <i>k</i> (coprime to <i>n</i>) are outer unless <span class="nowrap"><i>k</i> = ±1</span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Examples_of_automorphism_groups">Examples of automorphism groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=12" title="Edit section: Examples of automorphism groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="texhtml">D<sub>9</sub></span> has 18 <a href="/wiki/Inner_automorphism" title="Inner automorphism">inner automorphisms</a>. As 2D isometry group D<sub>9</sub>, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36 <a href="/wiki/Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer automorphisms</a>; e.g., multiplying angles of rotation by 2. </p><p><span class="texhtml">D<sub>10</sub></span> has 10 inner automorphisms. As 2D isometry group D<sub>10</sub>, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms; e.g., multiplying rotations by 3. </p><p>Compare the values 6 and 4 for <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a>, the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo <i>n</i></a> for <i>n</i> = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order). </p><p>The only values of <i>n</i> for which <i>φ</i>(<i>n</i>) = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely <span class="texhtml">D<sub>3</sub></span> (order 6), <span class="texhtml">D<sub>4</sub></span> (order 8), and <span class="texhtml">D<sub>6</sub></span> (order 12).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Inner_automorphism_group">Inner automorphism group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=13" title="Edit section: Inner automorphism group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The inner automorphism group of <span class="texhtml">D<sub><i>n</i></sub></span> is isomorphic to:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <ul><li><span class="texhtml">D<sub><i>n</i></sub></span> if <i>n</i> is odd;</li> <li><span class="texhtml">D<sub><i>n</i></sub> / Z<sub>2</sub></span> if <span class="texhtml"><i>n</i></span> is even (for <span class="texhtml"><i>n</i> = 2</span>, <span class="texhtml">D<sub><i>2</i></sub> / Z<sub>2</sub> = <i>1</i></span> ).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=14" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several important generalizations of the dihedral groups: </p> <ul><li>The <a href="/wiki/Infinite_dihedral_group" title="Infinite dihedral group">infinite dihedral group</a> is an <a href="/wiki/Infinite_group" title="Infinite group">infinite group</a> with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the <a href="/wiki/Integer" title="Integer">integers</a>.</li> <li>The <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(2), <i>i.e.,</i> the symmetry group of the <a href="/wiki/Circle" title="Circle">circle</a>, also has similar properties to the dihedral groups.</li> <li>The family of <a href="/wiki/Generalized_dihedral_group" title="Generalized dihedral group">generalized dihedral groups</a> includes both of the examples above, as well as many other groups.</li> <li>The <a href="/wiki/Quasidihedral_group" title="Quasidihedral group">quasidihedral groups</a> are family of finite groups with similar properties to the dihedral groups.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Dihedral_groups" class="extiw" title="commons:Category:Dihedral groups">Dihedral groups</a></span>.</div></div> </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/Cycle_index#Dihedral_group_Dn" title="Cycle index">Cycle index of the dihedral group</a></li> <li><a href="/wiki/Dicyclic_group" title="Dicyclic group">Dicyclic group</a></li> <li><a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">Dihedral group of order 6</a></li> <li><a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">Dihedral group of order 8</a></li> <li><a href="/wiki/Point_groups_in_three_dimensions#Symmetry_groups_in_3D_that_are_dihedral_as_abstract_group" title="Point groups in three dimensions">Dihedral symmetry groups in 3D</a></li> <li><a href="/wiki/Dihedral_symmetry_in_three_dimensions" title="Dihedral symmetry in three dimensions">Dihedral symmetry in three dimensions</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Dihedral_Group"><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="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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(1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2jBqvVb0Q-AC&pg=PA195"><i>A Course in Group Theory</i></a>. Oxford University Press. p. 195. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198534594" title="Special:BookSources/9780198534594"><bdi>9780198534594</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Group+Theory&rft.pages=195&rft.pub=Oxford+University+Press&rft.date=1996&rft.isbn=9780198534594&rft.aulast=Humphreys&rft.aufirst=John+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2jBqvVb0Q-AC%26pg%3DPA195&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPedersen" class="citation web cs1">Pedersen, John. <a rel="nofollow" class="external text" href="http://www.math.ucsd.edu/~atparris/small_groups.html">"Groups of small order"</a>. Dept of Mathematics, University of South Florida.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Groups+of+small+order&rft.pub=Dept+of+Mathematics%2C+University+of+South+Florida&rft.aulast=Pedersen&rft.aufirst=John&rft_id=http%3A%2F%2Fwww.math.ucsd.edu%2F~atparris%2Fsmall_groups.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSommer-Simpson2013" class="citation web cs1">Sommer-Simpson, Jasha (2 November 2013). <a rel="nofollow" class="external text" href="http://math.uchicago.edu/~may/REU2013/REUPapers/Sommer-Simpson.pdf">"Automorphism groups for semidirect products of cyclic groups"</a> <span class="cs1-format">(PDF)</span>. p. 13. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160806115458/http://math.uchicago.edu/~may/REU2013/REUPapers/Sommer-Simpson.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2016-08-06. <q><b>Corollary 7.3.</b> Aut(D<sub><i>n</i></sub>) = D<sub><i>n</i></sub> if and only if <i>φ</i>(<i>n</i>) = 2</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Automorphism+groups+for+semidirect+products+of+cyclic+groups&rft.pages=13&rft.date=2013-11-02&rft.aulast=Sommer-Simpson&rft.aufirst=Jasha&rft_id=http%3A%2F%2Fmath.uchicago.edu%2F~may%2FREU2013%2FREUPapers%2FSommer-Simpson.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller1942" class="citation journal cs1">Miller, GA (September 1942). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078492">"Automorphisms of the Dihedral Groups"</a>. <i>Proc Natl Acad Sci U S A</i>. <b>28</b> (9): 368–71. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1942PNAS...28..368M">1942PNAS...28..368M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.28.9.368">10.1073/pnas.28.9.368</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078492">1078492</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16588559">16588559</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc+Natl+Acad+Sci+U+S+A&rft.atitle=Automorphisms+of+the+Dihedral+Groups&rft.volume=28&rft.issue=9&rft.pages=368-71&rft.date=1942-09&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1078492%23id-name%3DPMC&rft_id=info%3Apmid%2F16588559&rft_id=info%3Adoi%2F10.1073%2Fpnas.28.9.368&rft_id=info%3Abibcode%2F1942PNAS...28..368M&rft.aulast=Miller&rft.aufirst=GA&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1078492&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_group&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130222032517/http://demonstrations.wolfram.com/DihedralGroupNOfOrder2n/">Dihedral Group n of Order 2n</a> by Shawn Dudzik, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li> <li><a rel="nofollow" class="external text" href="http://groupprops.subwiki.org/wiki/Dihedral_group">Dihedral group</a> at Groupprops</li> <li><span class="citation mathworld" id="Reference-Mathworld-Dihedral_Group"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DihedralGroup.html">"Dihedral Group"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Dihedral+Group&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDihedralGroup.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Dihedral_Group_D3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DihedralGroupD3.html">"Dihedral Group D3"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Dihedral+Group+D3&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDihedralGroupD3.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Dihedral_Group_D4"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DihedralGroupD4.html">"Dihedral Group D4"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Dihedral+Group+D4&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDihedralGroupD4.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Dihedral_Group_D5"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DihedralGroupD5.html">"Dihedral Group D5"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Dihedral+Group+D5&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDihedralGroupD5.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Dihedral_Group_D6"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Davis, Declan. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DihedralGroupD6.html">"Dihedral Group D6"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Dihedral+Group+D6&rft.au=Davis%2C+Declan&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDihedralGroupD6.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+group" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://groupnames.org/#?dihedral">Dihedral groups on GroupNames</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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colspan="2" style="background: #ceecee"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_navbox" title="Template:Group navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_navbox" title="Template talk:Group navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_navbox" title="Special:EditPage/Template:Group navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Groups" style="font-size:114%;margin:0 4em"><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Groups</a></div></th></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Group_theory" title="Group theory">Basic notions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <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/Commutator_subgroup" title="Commutator subgroup">Commutator subgroup</a></li> <li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphism</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></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%">Types of groups</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">Abelian groups</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic groups</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">Infinite group</a></li> <li><a href="/wiki/Simple_group" title="Simple group">Simple groups</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">Solvable groups</a></li> <li><a href="/wiki/Symmetry_group" title="Symmetry group">Symmetry group</a></li> <li><a href="/wiki/Space_group" title="Space group">Space group</a></li> <li><a href="/wiki/Point_group" title="Point group">Point group</a></li> <li><a href="/wiki/Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li> <li><a href="/wiki/Trivial_group" title="Trivial group">Trivial group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></dt> <dd><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub>n</sub></dd> <dd><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub>n</sub></dd></dl> <dl><dt><a href="/wiki/Sporadic_group" title="Sporadic group">Sporadic groups</a></dt> <dd><a href="/wiki/Mathieu_group" title="Mathieu group">Mathieu group</a> M<sub>11..12</sub>,M<sub>22..24</sub></dd> <dd><a href="/wiki/Conway_group" title="Conway group">Conway group</a> Co<sub>1..3</sub></dd> <dd>Janko groups <a href="/wiki/Janko_group_J1" title="Janko group J1">J<sub>1</sub></a>, <a href="/wiki/Janko_group_J2" title="Janko group J2">J<sub>2</sub></a>, <a href="/wiki/Janko_group_J3" title="Janko group J3">J<sub>3</sub></a>, <a href="/wiki/Janko_group_J4" title="Janko group J4">J<sub>4</sub></a></dd> <dd><a href="/wiki/Fischer_group" title="Fischer group">Fischer group</a> F<sub>22..24</sub></dd> <dd><a href="/wiki/Baby_monster_group" title="Baby monster group">Baby monster group</a> B</dd> <dd><a href="/wiki/Monster_group" title="Monster group">Monster group</a> M</dd></dl> <dl><dt>Other finite groups</dt> <dd><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> <i>S</i><sub><i>n</i></sub></dd> <dd><a class="mw-selflink selflink">Dihedral group</a> <i>D</i><sub><i>n</i></sub></dd> <dd><a href="/wiki/Rubik%27s_Cube_group" title="Rubik's Cube group">Rubik's Cube group</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Lie_group" title="Lie group">Lie groups</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear group</a> GL(n)</li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear group</a> SL(n)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal group</a> O(n)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Special orthogonal group</a> SO(n)</li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary group</a> U(n)</li> <li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary group</a> SU(n)</li> <li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic group</a> Sp(n)</li></ul> <dl><dt><a href="/wiki/Exceptional_Lie_groups" class="mw-redirect" title="Exceptional Lie groups">Exceptional Lie groups</a></dt> <dd><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></dd> <dd><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></dd> <dd><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></dd> <dd><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></dd> <dd><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></dd></dl> <ul><li><a href="/wiki/Circle_group" title="Circle group">Circle group</a></li> <li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%">Infinite dimensional groups</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism group</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop group</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li>O(∞)</li> <li>SU(∞)</li> <li>Sp(∞)</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="background: #ceecee;font-weight:bold;"><div> <ul><li><a href="/wiki/History_of_group_theory" title="History of group theory">History</a></li> <li><a href="/wiki/Group_theory#Applications_of_group_theory" title="Group theory">Applications</a></li> <li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> 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Dihedral group - Wikipedia