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id="toc-Real-valued_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real-valued_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Real-valued functions</span> </div> </a> <ul id="toc-Real-valued_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Euclidean geometry</span> </div> </a> <ul id="toc-Euclidean_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Projective geometry</span> </div> </a> <ul id="toc-Projective_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Linear algebra</span> </div> </a> <ul id="toc-Linear_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quaternion_algebra,_groups,_semigroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quaternion_algebra,_groups,_semigroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Quaternion algebra, groups, semigroups</span> </div> </a> <ul id="toc-Quaternion_algebra,_groups,_semigroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ring_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ring_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Ring theory</span> </div> </a> <ul id="toc-Ring_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Group theory</span> </div> </a> <ul id="toc-Group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Mathematical logic</span> </div> </a> <ul id="toc-Mathematical_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Computer science</span> </div> </a> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.10</span> <span>Physics</span> </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> </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">4</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">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">7</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 type="checkbox" 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Available in 32 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-32" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">32 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%A7%D8%B1%D8%AA%D8%AF%D8%A7%D8%AF%D9%8A%D8%A9" title="دالة ارتدادية – Arabic" lang="ar" hreflang="ar" data-title="دالة ارتدادية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Involuci%C3%B3" title="Involució – Catalan" lang="ca" hreflang="ca" data-title="Involució" 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/Involuce_(matematika)" title="Involuce (matematika) – Czech" lang="cs" hreflang="cs" data-title="Involuce (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Infolytedd" title="Infolytedd – Welsh" lang="cy" hreflang="cy" data-title="Infolytedd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Involution_(Mathematik)" title="Involution (Mathematik) – German" lang="de" hreflang="de" data-title="Involution (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Involutsioon_(matemaatika)" title="Involutsioon (matemaatika) – Estonian" lang="et" hreflang="et" data-title="Involutsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Involuci%C3%B3n_(matem%C3%A1tica)" title="Involución (matemática) – Spanish" lang="es" hreflang="es" data-title="Involución (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Involucio" title="Involucio – Esperanto" lang="eo" hreflang="eo" data-title="Involucio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%DB%8C%DA%86%D8%B4_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="پیچش (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="پیچش (ریاضیات)" 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/Involution_(math%C3%A9matiques)" title="Involution (mathématiques) – French" lang="fr" hreflang="fr" data-title="Involution (mathématiques)" 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/%EB%8C%80%ED%95%A9_(%EC%88%98%ED%95%99)" title="대합 (수학) – Korean" lang="ko" hreflang="ko" data-title="대합 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Involution" title="Involution – Interlingua" lang="ia" hreflang="ia" data-title="Involution" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sj%C3%A1lfhverfa" title="Sjálfhverfa – Icelandic" lang="is" hreflang="is" data-title="Sjálfhverfa" 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/Involuzione_(teoria_degli_insiemi)" title="Involuzione (teoria degli insiemi) – Italian" lang="it" hreflang="it" data-title="Involuzione (teoria degli insiemi)" 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%90%D7%99%D7%A0%D7%95%D7%95%D7%9C%D7%95%D7%A6%D7%99%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="אינוולוציה (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="אינוולוציה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Involutie_(wiskunde)" title="Involutie (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Involutie (wiskunde)" 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/%E5%AF%BE%E5%90%88" title="対合 – Japanese" lang="ja" hreflang="ja" data-title="対合" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Involusjon_i_matematikk" title="Involusjon i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Involusjon i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Anvolussion" title="Anvolussion – Piedmontese" lang="pms" hreflang="pms" data-title="Anvolussion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Inwolucja_(matematyka)" title="Inwolucja (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Inwolucja (matematyka)" 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/Involu%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Involução (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Involução (matemática)" 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/Involu%C8%9Bie_(matematic%C4%83)" title="Involuție (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Involuție (matematică)" 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%98%D0%BD%D0%B2%D0%BE%D0%BB%D1%8E%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Инволюция (математика) – Russian" lang="ru" hreflang="ru" data-title="Инволюция (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Invol%C3%BAcia_(matematika)" title="Involúcia (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Involúcia (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Involucija_(matematika)" title="Involucija (matematika) – Slovenian" lang="sl" hreflang="sl" data-title="Involucija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%BE%D0%BB%D1%83%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Инволуција (математика) – Serbian" lang="sr" hreflang="sr" data-title="Инволуција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Involution_(matematik)" title="Involution (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Involution (matematik)" 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%9A%E0%AF%81%E0%AE%B0%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B5%E0%AF%81_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="சுருள்வு (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="சுருள்வு (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%B2%E0%B8%A7%E0%B8%B1%E0%B8%95%E0%B8%99%E0%B8%B2%E0%B8%81%E0%B8%B2%E0%B8%A3" title="อาวัตนาการ – Thai" lang="th" hreflang="th" data-title="อาวัตนาการ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D0%B2%D0%BE%D0%BB%D1%8E%D1%86%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Інволюція (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Інволюція (математика)" 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/H%C3%A0m_s%E1%BB%91_t%E1%BB%B1_ngh%E1%BB%8Bch_%C4%91%E1%BA%A3o" title="Hàm số tự nghịch đảo – Vietnamese" lang="vi" hreflang="vi" data-title="Hàm số tự nghịch đảo" 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 href="https://zh.wikipedia.org/wiki/%E5%B0%8D%E5%90%88" title="對合 – Chinese" lang="zh" hreflang="zh" data-title="對合" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q846862#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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.hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Involution_(disambiguation)#Mathematics" class="mw-redirect mw-disambig" title="Involution (disambiguation)">Involution (disambiguation) § Mathematics</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Involution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Involution.svg/220px-Involution.svg.png" decoding="async" width="220" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Involution.svg/330px-Involution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Involution.svg/440px-Involution.svg.png 2x" data-file-width="788" data-file-height="400" /></a><figcaption>An involution is a function <span class="texhtml"><i>f</i> : <i>X</i> → <i>X</i></span> that, when applied twice, brings one back to the starting point.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>involution</b>, <b>involutory function</b>, or <b>self-inverse function</b><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> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml mvar" style="font-style:italic;">f</span> that is its own <a href="/wiki/Inverse_function" title="Inverse function">inverse</a>, </p> <dl><dd><span class="texhtml"><i>f</i>(<i>f</i>(<i>x</i>)) = <i>x</i></span></dd></dl> <p>for all <span class="texhtml mvar" style="font-style:italic;">x</span> in the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of <span class="texhtml"><i>f</i></span>.<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> Equivalently, applying <span class="texhtml mvar" style="font-style:italic;">f</span> twice produces the original value. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="General_properties">General properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=1" title="Edit section: General properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any involution is a <a href="/wiki/Bijection" title="Bijection">bijection</a>. </p><p>The <a href="/wiki/Identity_function" title="Identity function">identity map</a> is a trivial example of an involution. Examples of nontrivial involutions include <a href="/wiki/Additive_inverse" title="Additive inverse">negation</a> (<span class="texhtml"><i>x</i> ↦ −<i>x</i></span>), <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocation</a> (<span class="texhtml"><i>x</i> ↦ 1/<i>x</i></span>), and <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugation</a> (<span class="texhtml"><i>z</i> ↦ <span style="text-decoration:overline;"><i>z</i></span></span>) in <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>; <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>, half-turn <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>, and <a href="/wiki/Circle_inversion" class="mw-redirect" title="Circle inversion">circle inversion</a> in <a href="/wiki/Geometry" title="Geometry">geometry</a>; <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complementation</a> in <a href="/wiki/Set_theory" title="Set theory">set theory</a>; and <a href="/wiki/Reciprocal_cipher" class="mw-redirect" title="Reciprocal cipher">reciprocal ciphers</a> such as the <a href="/wiki/ROT13" title="ROT13">ROT13</a> transformation and the <a href="/wiki/Beaufort_cipher" title="Beaufort cipher">Beaufort</a> <a href="/wiki/Polyalphabetic_cipher" title="Polyalphabetic cipher">polyalphabetic cipher</a>. </p><p>The <a href="/wiki/Function_composition" title="Function composition">composition</a> <span class="texhtml"><i>g</i> ∘ <i>f</i></span> of two involutions <span class="texhtml"><i>f</i></span> and <span class="texhtml"><i>g</i></span> is an involution if and only if they <a href="/wiki/Commutative_property" title="Commutative property">commute</a>: <span class="texhtml"><i>g</i> ∘ <i>f</i> = <i>f</i> ∘ <i>g</i></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Involutions_on_finite_sets">Involutions on finite sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=2" title="Edit section: Involutions on finite sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The number of involutions, including the identity involution, on a set with <span class="texhtml"><i>n</i> = 0, 1, 2, ...</span> elements is given by a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> found by <a href="/wiki/Heinrich_August_Rothe" title="Heinrich August Rothe">Heinrich August Rothe</a> in 1800: </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 a_{0}=a_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=a_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7539f1e4c55563abb37f28f342624c7e768de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.927ex; height:2.509ex;" alt="{\displaystyle a_{0}=a_{1}=1}"></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 a_{n}=a_{n-1}+(n-1)a_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=a_{n-1}+(n-1)a_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70a39e9e817160deb8265a4f086321636862344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.692ex; height:2.843ex;" alt="{\displaystyle a_{n}=a_{n-1}+(n-1)a_{n-2}}"></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 n>1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590cc6dae7bc8470936e1f47e6df667458c7ea6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.176ex;" alt="{\displaystyle n>1.}"></span></dd></dl> <p>The first few terms of this sequence are <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, 1, <a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, <a href="/wiki/4_(number)" class="mw-redirect" title="4 (number)">4</a>, <a href="/wiki/10_(number)" class="mw-redirect" title="10 (number)">10</a>, <a href="/wiki/26_(number)" title="26 (number)">26</a>, <a href="/wiki/76_(number)" title="76 (number)">76</a>, <a href="/wiki/232_(number)" title="232 (number)">232</a> (sequence <span class="nowrap external"><a href="//oeis.org/A000085" class="extiw" title="oeis:A000085">A000085</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>); these numbers are called the <a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">telephone numbers</a>, and they also count the number of <a href="/wiki/Young_tableau" title="Young tableau">Young tableaux</a> with a given number of cells.<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> The number <span class="texhtml"><i>a</i><sub><i>n</i></sub></span> can also be expressed by non-recursive formulas, such as the sum <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">⌋</mo> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>m</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cefc2522c0735916844dc264b555aaf6cce79874" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.379ex; height:8.343ex;" alt="{\displaystyle a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.}"></span> </p><p>The number of fixed points of an involution on a finite set and its <a href="/wiki/Cardinality" title="Cardinality">number of elements</a> have the same <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">parity</a>. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an <a href="/wiki/Odd_number" class="mw-redirect" title="Odd number">odd number</a> of elements has at least one <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>. This can be used to prove <a href="/wiki/Fermat%27s_theorem_on_sums_of_two_squares#Zagier's_"one-sentence_proof"" title="Fermat's theorem on sums of two squares">Fermat's two squares theorem</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Involution_throughout_the_fields_of_mathematics">Involution throughout the fields of mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=3" title="Edit section: Involution throughout the fields of mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Real-valued_functions">Real-valued functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=4" title="Edit section: Real-valued functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of an involution (on the real numbers) is <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">symmetric</a> across the line <span class="texhtml"><i>y</i> = <i>x</i></span>. This is due to the fact that the inverse of any <i>general</i> function will be its reflection over the line <span class="texhtml"><i>y</i> = <i>x</i></span>. This can be seen by "swapping" <span class="texhtml mvar" style="font-style:italic;">x</span> with <span class="texhtml mvar" style="font-style:italic;">y</span>. If, in particular, the function is an <i>involution</i>, then its graph is its own reflection. Some basic examples of involutions include the functions <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{1}f_{1}(x)&=a-x,\\f_{2}(x)&={\frac {b}{x}},\\f_{3}(x)&={\frac {x}{cx-1}},\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="3pt" columnspacing="0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>x</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>c</mi> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{1}f_{1}(x)&=a-x,\\f_{2}(x)&={\frac {b}{x}},\\f_{3}(x)&={\frac {x}{cx-1}},\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4efcc1b6aebc02b22a4215fc48851c988213c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:17.005ex; height:13.509ex;" alt="{\displaystyle {\begin{alignedat}{1}f_{1}(x)&=a-x,\\f_{2}(x)&={\frac {b}{x}},\\f_{3}(x)&={\frac {x}{cx-1}},\\\end{alignedat}}}"></span> These may be composed in various ways to produce additional involutions. For example, if <span class="texhtml"><i>a</i>=0</span> and <span class="texhtml"><i>b</i>=1</span> then <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 f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d49f3fdc56c173daa30379f56387b778199cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.308ex; height:5.176ex;" alt="{\displaystyle f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}}}"></span> is an involution, and more generally the function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {x+b}{cx-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>c</mi> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {x+b}{cx-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acab17e9921dc4404a92ef84b11549a8929acdf1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.529ex; height:5.509ex;" alt="{\displaystyle g(x)={\frac {x+b}{cx-1}}}"></span> is an involution for constants <span class="texhtml mvar" style="font-style:italic;">b</span> and <span class="texhtml mvar" style="font-style:italic;">c</span> which satisfy <span class="texhtml"><i>bc</i> ≠ −1</span>. (This is the self-inverse subset of <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> with <span class="texhtml"><i>a</i> = −<i>d</i></span>, then normalized to <span class="texhtml"><i>a</i> = 1</span>.) </p><p>Other nonlinear examples can be constructed by wrapping an involution <span class="texhtml mvar" style="font-style:italic;">g</span> in an arbitrary function <span class="texhtml mvar" style="font-style:italic;">h</span> and its inverse, producing <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 f:=h^{-1}\circ g\circ h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:=h^{-1}\circ g\circ h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2c35e8242a07058855ce550f916f0da1cea825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.54ex; height:3.009ex;" alt="{\displaystyle f:=h^{-1}\circ g\circ h}"></span>, such as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}f(x)&={\sqrt {1-x^{2}}}&g(x)&=1-x&h(x)&=x^{2},\\f(x)&=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right)&g(x)&={\frac {x+1}{x-1}}&h(x)&=e^{x},\\f(x)&=\exp \left({\frac {1}{\ln x}}\right)&g(x)&={\frac {1}{x}}&h(x)&=\ln x,\\f(x)&={\frac {x}{\sqrt {x^{2}-1}}}&\qquad g(x)&={\frac {x}{x-1}}&\quad h(x)&=x^{2}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mtd> <mtd> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mspace width="2em" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}f(x)&={\sqrt {1-x^{2}}}&g(x)&=1-x&h(x)&=x^{2},\\f(x)&=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right)&g(x)&={\frac {x+1}{x-1}}&h(x)&=e^{x},\\f(x)&=\exp \left({\frac {1}{\ln x}}\right)&g(x)&={\frac {1}{x}}&h(x)&=\ln x,\\f(x)&={\frac {x}{\sqrt {x^{2}-1}}}&\qquad g(x)&={\frac {x}{x-1}}&\quad h(x)&=x^{2}.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c77bababaa530c7abe6428a31ccc1e3e62161b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:53.092ex; height:22.176ex;" alt="{\displaystyle {\begin{alignedat}{3}f(x)&={\sqrt {1-x^{2}}}&g(x)&=1-x&h(x)&=x^{2},\\f(x)&=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right)&g(x)&={\frac {x+1}{x-1}}&h(x)&=e^{x},\\f(x)&=\exp \left({\frac {1}{\ln x}}\right)&g(x)&={\frac {1}{x}}&h(x)&=\ln x,\\f(x)&={\frac {x}{\sqrt {x^{2}-1}}}&\qquad g(x)&={\frac {x}{x-1}}&\quad h(x)&=x^{2}.\end{alignedat}}}"></span> </p><p>Other elementary involutions are useful in <a href="/wiki/Functional_equation#Involutions" title="Functional equation">solving functional equations</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Euclidean_geometry">Euclidean geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=5" title="Edit section: Euclidean geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple example of an involution of the three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> is <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> through a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>. Performing a reflection twice brings a point back to its original coordinates. </p><p>Another involution is <a href="/wiki/Reflection_through_the_origin" class="mw-redirect" title="Reflection through the origin">reflection through the origin</a>; not a reflection in the above sense, and so, a distinct example. </p><p>These transformations are examples of <a href="/wiki/Affine_involution" title="Affine involution">affine involutions</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Projective_geometry">Projective geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=6" title="Edit section: Projective geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An involution is a <a href="/wiki/Projectivity" class="mw-redirect" title="Projectivity">projectivity</a> of period 2, that is, a projectivity that interchanges pairs of points.<sup id="cite_ref-AGP_6-0" class="reference"><a href="#cite_note-AGP-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 24">: 24 </span></sup> </p> <ul><li>Any projectivity that interchanges two points is an involution.</li> <li>The three pairs of opposite sides of a <a href="/wiki/Complete_quadrangle" title="Complete quadrangle">complete quadrangle</a> meet any line (not through a vertex) in three pairs of an involution. This theorem has been called <a href="/wiki/Desargues" class="mw-redirect" title="Desargues">Desargues</a>'s Involution Theorem.<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> Its origins can be seen in Lemma IV of the lemmas to the <i>Porisms</i> of Euclid in Volume VII of the <i>Collection</i> of <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a>.<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></li> <li>If an involution has one <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>, it has another, and consists of the correspondence between <a href="/wiki/Projective_harmonic_conjugate" title="Projective harmonic conjugate">harmonic conjugates</a> with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called <b>double points</b>.<sup id="cite_ref-AGP_6-1" class="reference"><a href="#cite_note-AGP-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 53">: 53 </span></sup></li></ul> <p>Another type of involution occurring in projective geometry is a <b>polarity</b> that is a <a href="/wiki/Correlation_(projective_geometry)" title="Correlation (projective geometry)">correlation</a> of period 2.<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="Linear_algebra">Linear algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=7" title="Edit section: Linear algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Involutory_matrix" title="Involutory matrix">Involutory matrix</a></div> <p>In linear algebra, an involution is a linear operator <span class="texhtml"><i>T</i></span> on a vector space, such that <span class="texhtml"><span style="padding-right:0.15em;"><i>T</i></span><sup>2</sup> = <i>I</i></span>. Except for in characteristic 2, such operators are diagonalizable for a given basis with just <span class="texhtml">1</span>s and <span class="texhtml">−1</span>s on the diagonal of the corresponding matrix. If the operator is orthogonal (an <b>orthogonal involution</b>), it is orthonormally diagonalizable. </p><p>For example, suppose that a basis for a vector space <span class="texhtml"><i>V</i></span> is chosen, and that <span class="texhtml"><i>e</i><sub>1</sub></span> and <span class="texhtml"><i>e</i><sub>2</sub></span> are basis elements. There exists a linear transformation <span class="texhtml"><i>f</i></span> that sends <span class="texhtml"><i>e</i><sub>1</sub></span> to <span class="texhtml"><i>e</i><sub>2</sub></span>, and sends <span class="texhtml"><i>e</i><sub>2</sub></span> to <span class="texhtml"><i>e</i><sub>1</sub></span>, and that is the identity on all other basis vectors. It can be checked that <span class="texhtml"><i>f</i>(<i>f</i>(<i>x</i>)) = <i>x</i></span> for all <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>V</i></span>. That is, <span class="texhtml"><i>f</i></span> is an involution of <span class="texhtml"><i>V</i></span>. </p><p>For a specific basis, any linear operator can be represented by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> <span class="texhtml"><i>T</i></span>. Every matrix has a <a href="/wiki/Transpose" title="Transpose">transpose</a>, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a> is an independent involution, the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> or <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian adjoint</a> is also an involution. </p><p>The definition of involution extends readily to <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>. Given a module <span class="texhtml"><i>M</i></span> over a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <span class="texhtml"><i>R</i></span>, an <span class="texhtml"><i>R</i></span> <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> <span class="texhtml"><i>f</i></span> of <span class="texhtml"><i>M</i></span> is called an involution if <span class="texhtml"><span style="padding-right:0.15em;"><i>f</i></span><sup>2</sup></span> is the identity homomorphism on <span class="texhtml"><i>M</i></span>. </p><p><a href="/wiki/Idempotent_element_(ring_theory)#Relation_with_involutions" class="mw-redirect" title="Idempotent element (ring theory)">Involutions are related to idempotents</a>; if <span class="texhtml">2</span> is invertible then they <a href="/wiki/Bijection" title="Bijection">correspond</a> in a one-to-one manner. </p><p>In <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, <a href="/wiki/Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">Banach *-algebras</a> and <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a> are special types of <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a> with involutions. </p> <div class="mw-heading mw-heading3"><h3 id="Quaternion_algebra,_groups,_semigroups"><span id="Quaternion_algebra.2C_groups.2C_semigroups"></span>Quaternion algebra, groups, semigroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=8" title="Edit section: Quaternion algebra, groups, semigroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Quaternion_algebra" title="Quaternion algebra">quaternion algebra</a>, an (anti-)involution is defined by the following axioms: if we consider a transformation <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 x\mapsto f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56846d99e780ebee609728e76389fa84f6cbac81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.361ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(x)}"></span> then it is an involution if </p> <ul><li><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 f(f(x))=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(f(x))=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04c2f87b1313815142f4089bddf9881977e4bdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.934ex; height:2.843ex;" alt="{\displaystyle f(f(x))=x}"></span> (it is its own inverse)</li> <li><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 f(x_{1}+x_{2})=f(x_{1})+f(x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1}+x_{2})=f(x_{1})+f(x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e6622a3152f52e4d672d641e8228d2665cb1a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.579ex; height:2.843ex;" alt="{\displaystyle f(x_{1}+x_{2})=f(x_{1})+f(x_{2})}"></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 f(\lambda x)=\lambda f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\lambda x)=\lambda f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b42352460cf33e32f4eaed1b01856c9627a235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.644ex; height:2.843ex;" alt="{\displaystyle f(\lambda x)=\lambda f(x)}"></span> (it is linear)</li> <li><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 f(x_{1}x_{2})=f(x_{1})f(x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1}x_{2})=f(x_{1})f(x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89df8eab3b7f35eeb7139f8517500b639bce075b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.898ex; height:2.843ex;" alt="{\displaystyle f(x_{1}x_{2})=f(x_{1})f(x_{2})}"></span></li></ul> <p>An anti-involution does not obey the last axiom but instead </p> <ul><li><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 f(x_{1}x_{2})=f(x_{2})f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1}x_{2})=f(x_{2})f(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3dc332aac7418a92b9f546376eeb623ff15a9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.898ex; height:2.843ex;" alt="{\displaystyle f(x_{1}x_{2})=f(x_{2})f(x_{1})}"></span></li></ul> <p>This former law is sometimes called <a href="/wiki/Antidistributive" class="mw-redirect" title="Antidistributive">antidistributive</a>. It also appears in <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> as <span class="texhtml">(<i>xy</i>)<sup>−1</sup> = (<i>y</i>)<sup>−1</sup>(<i>x</i>)<sup>−1</sup></span>. Taken as an axiom, it leads to the notion of <a href="/wiki/Semigroup_with_involution" title="Semigroup with involution">semigroup with involution</a>, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the <a href="/wiki/Full_linear_monoid" class="mw-redirect" title="Full linear monoid">full linear monoid</a>) with <a href="/wiki/Transpose" title="Transpose">transpose</a> as the involution. </p> <div class="mw-heading mw-heading3"><h3 id="Ring_theory">Ring theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=9" title="Edit section: Ring theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/*-algebra" title="*-algebra">*-algebra</a></div> <p>In <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>, the word <i>involution</i> is customarily taken to mean an <a href="/wiki/Antihomomorphism" title="Antihomomorphism">antihomomorphism</a> that is its own inverse function. Examples of involutions in common rings: </p> <ul><li><a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a> on the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, and its equivalent in the <a href="/wiki/Split-complex_number" title="Split-complex number">split-complex numbers</a></li> <li>taking the transpose in a matrix ring.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Group_theory">Group theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=10" title="Edit section: Group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Group_theory" title="Group theory">group theory</a>, an element of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is an involution if it has <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> 2; that is, an involution is an element <span class="texhtml"><i>a</i></span> such that <span class="texhtml"><i>a</i> ≠ <i>e</i></span> and <span class="texhtml"><i>a</i><sup>2</sup> = <i>e</i></span>, where <span class="texhtml"><i>e</i></span> is the <a href="/wiki/Identity_element" title="Identity element">identity element</a>.<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> Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is, <i>group</i> was taken to mean <i><a href="/wiki/Permutation_group" title="Permutation group">permutation group</a></i>. By the end of the 19th century, <i>group</i> was defined more broadly, and accordingly so was <i>involution</i>. </p><p>A <a href="/wiki/Permutation" title="Permutation">permutation</a> is an involution if and only if it can be written as a finite product of disjoint <a href="/wiki/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">transpositions</a>. </p><p>The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>. </p><p>An element <span class="texhtml"><i>x</i></span> of a group <span class="texhtml"><i>G</i></span> is called <a href="/wiki/Strongly_real_element" class="mw-redirect" title="Strongly real element">strongly real</a> if there is an involution <span class="texhtml"><i>t</i></span> with <span class="texhtml"><i>x</i><sup><i>t</i></sup> = <i>x</i><sup>−1</sup></span> (where <span class="texhtml"><i>x</i><sup><i>t</i></sup> = <i>x</i><sup>−1</sup> = <i>t</i><sup>−1</sup> ⋅ <i>x</i> ⋅ <i>t</i></span>). </p><p><a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a> are groups generated by a set <span class="texhtml"><i>S</i></span> of involutions subject only to relations involving powers of pairs of elements of <span class="texhtml"><i>S</i></span>. Coxeter groups can be used, among other things, to describe the possible <a href="/wiki/Platonic_solid" title="Platonic solid">regular polyhedra</a> and their <a href="/wiki/Regular_polytope" title="Regular polytope">generalizations to higher dimensions</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_logic">Mathematical logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=11" title="Edit section: Mathematical logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The operation of complement in <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a> is an involution. Accordingly, <a href="/wiki/Negation" title="Negation">negation</a> in <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a> satisfies the <i><a href="/wiki/Double_negation" title="Double negation">law of double negation</a></i>: <span class="texhtml">¬¬<i>A</i></span> is equivalent to <span class="texhtml"><i>A</i></span>. </p><p>Generally in <a href="/wiki/Non-classical_logic" title="Non-classical logic">non-classical logics</a>, negation that satisfies the law of double negation is called <i>involutive</i>. In <a href="/wiki/Algebraic_semantics_(mathematical_logic)" title="Algebraic semantics (mathematical logic)">algebraic semantics</a>, such a negation is realized as an involution on the algebra of <a href="/wiki/Truth_value" title="Truth value">truth values</a>. Examples of logics that have involutive negation are Kleene and Bochvar <a href="/wiki/Three-valued_logic" title="Three-valued logic">three-valued logics</a>, <a href="/wiki/%C5%81ukasiewicz_logic" title="Łukasiewicz logic">Łukasiewicz many-valued logic</a>, the <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a> '<a href="/wiki/Monoidal_t-norm_logic" title="Monoidal t-norm logic">involutive monoidal t-norm logic</a>' (IMTL), etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in <a href="/wiki/T-norm_fuzzy_logics" title="T-norm fuzzy logics">t-norm fuzzy logics</a>. </p><p>The involutiveness of negation is an important characterization property for logics and the corresponding <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">varieties of algebras</a>. For instance, involutive negation characterizes <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a> among <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebras</a>. Correspondingly, classical <a href="/wiki/Classical_logic" title="Classical logic">Boolean logic</a> arises by adding the law of double negation to <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. The same relationship holds also between <a href="/wiki/MV-algebra" title="MV-algebra">MV-algebras</a> and <a href="/wiki/BL_(logic)" title="BL (logic)">BL-algebras</a> (and so correspondingly between <a href="/wiki/%C5%81ukasiewicz_logic" title="Łukasiewicz logic">Łukasiewicz logic</a> and fuzzy logic <a href="/wiki/BL_(logic)" title="BL (logic)">BL</a>), IMTL and <a href="/wiki/Monoidal_t-norm_logic" title="Monoidal t-norm logic">MTL</a>, and other pairs of important varieties of algebras (respectively, corresponding logics). </p><p>In the study of <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>, every relation has a <a href="/wiki/Converse_relation" title="Converse relation">converse relation</a>. Since the converse of the converse is the original relation, the conversion operation is an involution on the <a href="/wiki/Category_of_relations" title="Category of relations">category of relations</a>. Binary relations are <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">ordered</a> through <a href="/wiki/Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">inclusion</a>. While this ordering is reversed with the <a href="/wiki/Complementation_(mathematics)" class="mw-redirect" title="Complementation (mathematics)">complementation</a> involution, it is preserved under conversion. </p> <div class="mw-heading mw-heading3"><h3 id="Computer_science">Computer science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=12" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/XOR" class="mw-redirect" title="XOR">XOR</a> <a href="/wiki/Bitwise_operation" title="Bitwise operation">bitwise operation</a> with a given value for one parameter is an involution on the other parameter. XOR <a href="/wiki/Mask_(computing)" title="Mask (computing)">masks</a> in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state. </p><p>Two special cases of this, which are also involutions, are the <a href="/wiki/Bitwise_NOT" class="mw-redirect" title="Bitwise NOT">bitwise NOT</a> operation which is XOR with an all-ones value, and <a href="/wiki/Stream_cipher" title="Stream cipher">stream cipher</a> <a href="/wiki/Encryption" title="Encryption">encryption</a>, which is an XOR with a secret <a href="/wiki/Keystream" title="Keystream">keystream</a>. </p><p>This predates binary computers; practically all mechanical cipher machines implement a <a href="/wiki/Reciprocal_cipher" class="mw-redirect" title="Reciprocal cipher">reciprocal cipher</a>, an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Another involution used in computers is an order-2 bitwise permutation. For example. a color value stored as integers in the form <span class="texhtml">(<i>R</i>, <i>G</i>, <i>B</i>)</span>, could exchange <span class="texhtml"><i>R</i></span> and <span class="texhtml"><i>B</i></span>, resulting in the form <span class="texhtml">(<i>B</i>, <i>G</i>, <i>R</i>)</span>: <span class="texhtml"><i>f</i>(<i>f</i>(RGB)) = RGB, <i>f</i>(<i>f</i>(BGR)) = BGR</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=13" title="Edit section: Physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a>, which converts between the <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> and <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a>, is an involutive operation. </p><p><a href="/wiki/Integrability" class="mw-disambig" title="Integrability">Integrability</a>, a central notion of physics and in particular the subfield of <a href="/wiki/Integrable_system" title="Integrable system">integrable systems</a>, is closely related to involution, for example in context of <a href="/wiki/Kramers%E2%80%93Wannier_duality" title="Kramers–Wannier duality">Kramers–Wannier duality</a>. </p> <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=Involution_(mathematics)&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Atbash" title="Atbash">Atbash</a></li> <li><a href="/wiki/Automorphism" title="Automorphism">Automorphism</a></li> <li><a href="/wiki/Idempotence" title="Idempotence">Idempotence</a></li> <li><a href="/wiki/ROT13" title="ROT13">ROT13</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=Involution_(mathematics)&action=edit&section=15" 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"> <div class="mw-references-wrap mw-references-columns"><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">Robert Alexander Adams, <i>Calculus: Single Variable</i>, 2006, <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0321307143" title="Special:BookSources/0321307143">0321307143</a>, p. 165</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRussell1903" class="citation cs2">Russell, Bertrand (1903), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=63ooitcP2osC&q=involution%20subject%3A%&pg=PR3"><i>Principles of mathematics</i></a> (2nd ed.), W. W. Norton & Company, Inc, p. 426, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781440054167" title="Special:BookSources/9781440054167"><bdi>9781440054167</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+mathematics&rft.pages=426&rft.edition=2nd&rft.pub=W.+W.+Norton+%26+Company%2C+Inc&rft.date=1903&rft.isbn=9781440054167&rft.aulast=Russell&rft.aufirst=Bertrand&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D63ooitcP2osC%26q%3Dinvolution%2520subject%253A%25%26pg%3DPR3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKubrusly2011" class="citation cs2">Kubrusly, Carlos S. (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g-UFYTO8SbMC&pg=PA27"><i>The Elements of Operator Theory</i></a>, Springer Science & Business Media, Problem 1.11(a), p. 27, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780817649982" title="Special:BookSources/9780817649982"><bdi>9780817649982</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Elements+of+Operator+Theory&rft.pages=Problem-1.11%28a%29%2C+p.-27&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=9780817649982&rft.aulast=Kubrusly&rft.aufirst=Carlos+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg-UFYTO8SbMC%26pg%3DPA27&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1973" class="citation cs2"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (1973), <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>, Volume 3: Sorting and Searching</i>, Reading, Mass.: Addison-Wesley, pp. 48, 65, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0445948">0445948</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming%2C+Volume+3%3A+Sorting+and+Searching&rft.place=Reading%2C+Mass.&rft.pages=48%2C+65&rft.pub=Addison-Wesley&rft.date=1973&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0445948%23id-name%3DMR&rft.aulast=Knuth&rft.aufirst=Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZagier1990" class="citation cs2"><a href="/wiki/Don_Zagier" title="Don Zagier">Zagier, D.</a> (1990), "A one-sentence proof that every prime <i>p</i> ≡ 1 (mod 4) is a sum of two squares", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>97</b> (2): 144, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2323918">10.2307/2323918</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2323918">2323918</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893">1041893</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+one-sentence+proof+that+every+prime+p+%E2%89%A1+1+%28mod+4%29+is+a+sum+of+two+squares&rft.volume=97&rft.issue=2&rft.pages=144&rft.date=1990&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1041893%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2323918%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2323918&rft.aulast=Zagier&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-AGP-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-AGP_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-AGP_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A.G. Pickford (1909) <a rel="nofollow" class="external text" href="https://archive.org/details/elementaryprojec00pickrich/page/n5">Elementary Projective Geometry</a>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/wiki/Judith_V._Field" title="Judith V. Field">J. V. Field</a> and J. J. Gray (1987) <i>The Geometrical Work of Girard Desargues</i>, (New York: Springer), p. 54</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">Ivor Thomas (editor) (1980) <i>Selections Illustrating the History of Greek Mathematics</i>, Volume II, number 362 in the <a href="/wiki/Loeb_Classical_Library" title="Loeb Classical Library">Loeb Classical Library</a> (Cambridge and London: Harvard and Heinemann), pp. 610–3</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"><a href="/wiki/H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">H. S. M. Coxeter</a> (1969) <i>Introduction to Geometry</i>, pp. 244–8, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a></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"> John S. Rose. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=j-I7Zpq3GdIC">"A Course on Group Theory"</a>. p. 10, section 1.13.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoebel2018" class="citation book cs1">Goebel, Greg (2018). <a rel="nofollow" class="external text" href="http://vc.airvectors.net/ttcode_05.html">"The Mechanization of Ciphers"</a>. <i>Classical Cryptology</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Mechanization+of+Ciphers&rft.btitle=Classical+Cryptology&rft.date=2018&rft.aulast=Goebel&rft.aufirst=Greg&rft_id=http%3A%2F%2Fvc.airvectors.net%2Fttcode_05.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Involution_(mathematics)&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEllSangwine2007" class="citation journal cs1">Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions". <i>Computers & Mathematics with Applications</i>. <b>53</b> (1): 137–143. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0506034">math/0506034</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.camwa.2006.10.029">10.1016/j.camwa.2006.10.029</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:45639619">45639619</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+%26+Mathematics+with+Applications&rft.atitle=Quaternion+involutions+and+anti-involutions&rft.volume=53&rft.issue=1&rft.pages=137-143&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0506034&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45639619%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.camwa.2006.10.029&rft.aulast=Ell&rft.aufirst=Todd+A.&rft.au=Sangwine%2C+Stephen+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnusMerkurjevRostTignol1998" class="citation cs2">Knus, Max-Albert; <a href="/wiki/Alexander_Merkurjev" title="Alexander Merkurjev">Merkurjev, Alexander</a>; <a href="/wiki/Markus_Rost" title="Markus Rost">Rost, Markus</a>; <a href="/wiki/Jean-Pierre_Tignol" title="Jean-Pierre Tignol">Tignol, Jean-Pierre</a> (1998), <i>The book of involutions</i>, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0904-0" title="Special:BookSources/0-8218-0904-0"><bdi>0-8218-0904-0</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0955.16001">0955.16001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+book+of+involutions&rft.place=Providence%2C+RI&rft.series=Colloquium+Publications&rft.pub=American+Mathematical+Society&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0955.16001%23id-name%3DZbl&rft.isbn=0-8218-0904-0&rft.aulast=Knus&rft.aufirst=Max-Albert&rft.au=Merkurjev%2C+Alexander&rft.au=Rost%2C+Markus&rft.au=Tignol%2C+Jean-Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Involution">"Involution"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Involution&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DInvolution&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvolution+%28mathematics%29" class="Z3988"></span></li></ul> <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=Involution_(mathematics)&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Involution" class="extiw" title="commons:Category:Involution">Involution</a> at Wikimedia Commons</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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