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multiplication</span> </div> </a> <ul id="toc-Modular_multiplication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotational_symmetries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotational_symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Rotational symmetries</span> </div> </a> <ul id="toc-Rotational_symmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Galois_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Galois_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Galois theory</span> </div> </a> <ul id="toc-Galois_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subgroups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Subgroups</span> </div> </a> <ul id="toc-Subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Additional_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Additional properties</span> </div> </a> <ul id="toc-Additional_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Associated_objects" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Associated_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Associated objects</span> </div> </a> <button aria-controls="toc-Associated_objects-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 Associated objects subsection</span> </button> <ul id="toc-Associated_objects-sublist" class="vector-toc-list"> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Representations</span> </div> </a> <ul id="toc-Representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cycle_graph" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cycle_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Cycle graph</span> </div> </a> <ul id="toc-Cycle_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cayley_graph" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cayley_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Cayley graph</span> </div> </a> <ul id="toc-Cayley_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Endomorphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Endomorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Endomorphisms</span> </div> </a> <ul id="toc-Endomorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_product_and_Hom_of_cyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_product_and_Hom_of_cyclic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Tensor product and Hom of cyclic groups</span> </div> </a> <ul id="toc-Tensor_product_and_Hom_of_cyclic_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_classes_of_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_classes_of_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Related classes of groups</span> </div> </a> <button aria-controls="toc-Related_classes_of_groups-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 Related classes of groups subsection</span> </button> <ul id="toc-Related_classes_of_groups-sublist" class="vector-toc-list"> <li id="toc-Virtually_cyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Virtually_cyclic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Virtually cyclic groups</span> </div> </a> <ul id="toc-Virtually_cyclic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Procyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Procyclic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Procyclic groups</span> </div> </a> <ul id="toc-Procyclic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Locally_cyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Locally_cyclic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Locally cyclic groups</span> </div> </a> <ul id="toc-Locally_cyclic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyclically_ordered_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclically_ordered_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Cyclically ordered groups</span> </div> </a> <ul id="toc-Cyclically_ordered_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metacyclic_and_polycyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metacyclic_and_polycyclic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Metacyclic and polycyclic groups</span> </div> </a> <ul id="toc-Metacyclic_and_polycyclic_groups-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">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Footnotes</span> </div> </a> <button aria-controls="toc-Footnotes-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 Footnotes subsection</span> </button> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> </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">9</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">10</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">11</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" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Cyclic group</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 31 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-31" 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">31 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%B2%D9%85%D8%B1%D8%A9_%D8%AF%D9%88%D8%B1%D9%8A%D8%A9" title="زمرة دورية – Arabic" lang="ar" hreflang="ar" data-title="زمرة دورية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D1%8B%D0%BA%D0%BB%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Цыклічная група – Belarusian" lang="be" hreflang="be" data-title="Цыклічная група" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_c%C3%ADclic" title="Grup cíclic – Catalan" lang="ca" hreflang="ca" data-title="Grup cíclic" 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/Cyklick%C3%A1_grupa" title="Cyklická grupa – Czech" lang="cs" hreflang="cs" data-title="Cyklická 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/Zyklische_Gruppe" title="Zyklische Gruppe – German" lang="de" hreflang="de" data-title="Zyklische Gruppe" 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_c%C3%ADclico" title="Grupo cíclico – Spanish" lang="es" hreflang="es" data-title="Grupo cíclico" 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/Cikla_grupo" title="Cikla grupo – Esperanto" lang="eo" hreflang="eo" data-title="Cikla grupo" 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/%DA%AF%D8%B1%D9%88%D9%87_%D8%AF%D9%88%D8%B1%DB%8C" title="گروه دوری – Persian" lang="fa" hreflang="fa" data-title="گروه دوری" 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_cyclique" title="Groupe cyclique – French" lang="fr" hreflang="fr" data-title="Groupe cyclique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Grupo_c%C3%ADclico" title="Grupo cíclico – Galician" lang="gl" hreflang="gl" data-title="Grupo cíclico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%9C%ED%99%98%EA%B5%B0" title="순환군 – Korean" lang="ko" hreflang="ko" data-title="순환군" 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_siklik" title="Grup siklik – Indonesian" lang="id" hreflang="id" data-title="Grup siklik" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_ciclico" title="Gruppo ciclico – Italian" lang="it" hreflang="it" data-title="Gruppo ciclico" 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%A6%D7%99%D7%A7%D7%9C%D7%99%D7%AA" title="חבורה ציקלית – Hebrew" lang="he" hreflang="he" data-title="חבורה ציקלית" 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/Ciklikus_csoport" title="Ciklikus csoport – Hungarian" lang="hu" hreflang="hu" data-title="Ciklikus csoport" 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%9A%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%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="ചാക്രികഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ചാക്രികഗ്രൂപ്പ്" 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/Cyclische_groep" title="Cyclische groep – Dutch" lang="nl" hreflang="nl" data-title="Cyclische 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/%E5%B7%A1%E5%9B%9E%E7%BE%A4" title="巡回群 – Japanese" lang="ja" hreflang="ja" data-title="巡回群" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Syklisk_gruppe" title="Syklisk gruppe – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Syklisk gruppe" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_cykliczna" title="Grupa cykliczna – Polish" lang="pl" hreflang="pl" data-title="Grupa cykliczna" 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_c%C3%ADclico" title="Grupo cíclico – Portuguese" lang="pt" hreflang="pt" data-title="Grupo cíclico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Циклическая группа – Russian" lang="ru" hreflang="ru" data-title="Циклическая группа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Циклична група – Serbian" lang="sr" hreflang="sr" data-title="Циклична група" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Cikli%C4%8Dna_grupa" title="Ciklična grupa – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Ciklična grupa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Syklinen_ryhm%C3%A4" title="Syklinen ryhmä – Finnish" lang="fi" hreflang="fi" data-title="Syklinen ryhmä" 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/Cyklisk_grupp" title="Cyklisk grupp – Swedish" lang="sv" hreflang="sv" data-title="Cyklisk 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%9A%E0%AF%81%E0%AE%B4%E0%AE%B1%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="சுழற் குலம் – Tamil" lang="ta" hreflang="ta" data-title="சுழற் குலம்" 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%A6%D0%B8%D0%BA%D0%BB%D1%96%D1%87%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Циклічна група – Ukrainian" lang="uk" hreflang="uk" data-title="Циклічна 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_cyclic" title="Nhóm cyclic – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm cyclic" 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searchaux" style="display:none">Mathematical group that can be generated as the set of powers of a single element</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><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></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 class="mw-selflink selflink">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><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 class="mw-selflink selflink">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 href="/wiki/Dihedral_group" title="Dihedral group">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> <p>In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, a <b>cyclic group</b> or <b>monogenous group</b> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, denoted C<sub><i>n</i></sub> (also frequently <b><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></b><sub><i>n</i></sub> or Z<sub><i>n</i></sub>, not to be confused with the commutative ring of <a href="/wiki/P-adic_number" title="P-adic number"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic numbers</a>), that is <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by a single element.<sup id="cite_ref-eom_1-0" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> That is, it is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Inverse_element" title="Inverse element">invertible</a> elements with a single <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a>, and it contains an element <i>g</i> such that every other element of the group may be obtained by repeatedly applying the group operation to <i>g</i> or its inverse. Each element can be written as an integer <a href="/wiki/Exponentiation" title="Exponentiation">power</a> of <i>g</i> in multiplicative notation, or as an integer multiple of <i>g</i> in additive notation. This element <i>g</i> is called a <i><a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generator</a></i> of the group.<sup id="cite_ref-eom_1-1" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Every infinite cyclic group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the <a href="/wiki/Additive_group" title="Additive group">additive group</a> of <b>Z</b>, the <a href="/wiki/Integer" title="Integer">integers</a>. Every finite cyclic group of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> <i>n</i> is isomorphic to the additive group of <a href="/wiki/Quotient_group" title="Quotient group"><b>Z</b>/<i>n</i><b>Z</b></a>, the integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <i>n</i>. Every cyclic group is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> (meaning that its group operation is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>), and every <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a> abelian group is a <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of cyclic groups. </p><p>Every cyclic group of <a href="/wiki/Prime_number" title="Prime number">prime</a> order is a <a href="/wiki/Simple_group" title="Simple group">simple group</a>, which cannot be broken down into smaller groups. In the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_notation">Definition and notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=1" title="Edit section: Definition and notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/160px-Cyclic_group.svg.png" decoding="async" width="160" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/320px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a><figcaption>The six 6th complex <a href="/wiki/Root_of_unity" title="Root of unity">roots of unity</a> form a cyclic group under multiplication. Here, <i>z</i> is a generator, but <i>z</i><sup>2</sup> is not, because its powers fail to produce the odd powers of <i>z</i>.</figcaption></figure> <p>For any element <i>g</i> in any group <i>G</i>, one can form the <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> that consists of all its integer <a href="/wiki/Exponentiation" title="Exponentiation">powers</a>: <span class="nowrap">⟨<i>g</i>⟩ = { <i>g</i><sup><i>k</i></sup> | <i>k</i> ∈ <b>Z</b> }</span>, called the <b>cyclic subgroup</b> generated by <i>g</i>. The <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of <i>g</i> is |⟨<i>g</i>⟩|, the number of elements in ⟨<i>g</i>⟩, conventionally abbreviated as |<i>g</i>|, as ord(<i>g</i>), or as o(<i>g</i>). That is, the order of an element is equal to the order of the cyclic subgroup that it generates. </p><p>A <i>cyclic group</i> is a group which is equal to one of its cyclic subgroups: <span class="nowrap"><i>G</i> = ⟨<i>g</i>⟩</span> for some element <i>g</i>, called a <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group"><i>generator</i></a> of <i>G</i>. </p><p>For a <b>finite cyclic group</b> <i>G</i> of order <i>n</i> we have <span class="nowrap"><i>G</i> = {<i>e</i>, <i>g</i>, <i>g</i><sup>2</sup>, ... , <i>g</i><sup><i>n</i>−1</sup>}</span>, where <i>e</i> is the identity element and <span class="nowrap"><i>g</i><sup><i>i</i></sup> = <i>g</i><sup><i>j</i></sup></span> whenever <span class="nowrap"><i>i</i> ≡ <i>j</i></span> (<a href="/wiki/Modular_arithmetic" title="Modular arithmetic">mod</a> <i>n</i>); in particular <span class="nowrap"><i>g</i><sup><i>n</i></sup> = <i>g</i><sup>0</sup> = <i>e</i></span>, and <span class="nowrap"><i>g</i><sup>−1</sup> = <i>g</i><sup><i>n</i>−1</sup></span>. An abstract group defined by this multiplication is often denoted C<sub><i>n</i></sub>, and we say that <i>G</i> is <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to the standard cyclic group C<sub><i>n</i></sub>. Such a group is also isomorphic to <b>Z</b>/<i>n</i><b>Z</b>, the group of integers modulo <i>n</i> with the addition operation, which is the standard cyclic group in additive notation. Under the isomorphism <i>χ</i> defined by <span class="nowrap"><i>χ</i>(<i>g</i><sup><i>i</i></sup>) = <i>i</i></span> the identity element <i>e</i> corresponds to 0, products correspond to sums, and powers correspond to multiples. </p><p>For example, the set of complex 6th roots of unity: <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=\left\{\pm 1,\pm {\left({\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i\right)},\pm {\left({\tfrac {1}{2}}-{\tfrac {\sqrt {3}}{2}}i\right)}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\left\{\pm 1,\pm {\left({\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i\right)},\pm {\left({\tfrac {1}{2}}-{\tfrac {\sqrt {3}}{2}}i\right)}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67147eb422e430ae75997cf9ef673d42ba2bfb0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.887ex; height:4.843ex;" alt="{\displaystyle G=\left\{\pm 1,\pm {\left({\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i\right)},\pm {\left({\tfrac {1}{2}}-{\tfrac {\sqrt {3}}{2}}i\right)}\right\}}"></span> forms a group under multiplication. It is cyclic, since it is generated by the <a href="/wiki/Root_of_unity#General_definition" title="Root of unity">primitive root</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 z={\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i=e^{2\pi i/6}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i=e^{2\pi i/6}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749b92028f9916dcdce4a2cd1c59823aa20d2d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.196ex; height:4.176ex;" alt="{\displaystyle z={\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i=e^{2\pi i/6}:}"></span> that is, <i>G</i> = ⟨<i>z</i>⟩ = { 1, <i>z</i>, <i>z</i><sup>2</sup>, <i>z</i><sup>3</sup>, <i>z</i><sup>4</sup>, <i>z</i><sup>5</sup> } with <i>z</i><sup>6</sup> = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C<sub>6</sub> = ⟨<i>g</i>⟩ = { <i>e</i>, <i>g</i>, <i>g</i><sup>2</sup>, <i>g</i><sup>3</sup>, <i>g</i><sup>4</sup>, <i>g</i><sup>5</sup> } with multiplication <i>g</i><sup><i>j</i></sup> · <i>g</i><sup><i>k</i></sup> = <i>g</i><sup><i>j</i>+<i>k</i></sup> <sup>(mod 6)</sup>, so that <i>g</i><sup>6</sup> = <i>g</i><sup>0</sup> = <i>e</i>. These groups are also isomorphic to <b>Z</b>/6<b>Z</b> = {0, 1, 2, 3, 4, 5} with the operation of addition <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 6, with <i>z</i><sup><i>k</i></sup> and <i>g</i><sup><i>k</i></sup> corresponding to <i>k</i>. For example, <span class="nowrap">1 + 2 ≡ 3 (mod 6)</span> corresponds to <span class="nowrap"><i>z</i><sup>1</sup> · <i>z</i><sup>2</sup> = <i>z</i><sup>3</sup></span>, and <span class="nowrap">2 + 5 ≡ 1 (mod 6)</span> corresponds to <span class="nowrap"><i>z</i><sup>2</sup> · <i>z</i><sup>5</sup> = <i>z</i><sup>7</sup> = <i>z</i><sup>1</sup></span>, and so on. Any element generates its own cyclic subgroup, such as ⟨<i>z</i><sup>2</sup>⟩ = { <i>e</i>, <i>z</i><sup>2</sup>, <i>z</i><sup>4</sup> } of order 3, isomorphic to C<sub>3</sub> and <b>Z</b>/3<b>Z</b>; and ⟨<i>z</i><sup>5</sup>⟩ = { <i>e</i>, <i>z</i><sup>5</sup>, <i>z</i><sup>10</sup> = <i>z</i><sup>4</sup>, <i>z</i><sup>15</sup> = <i>z</i><sup>3</sup>, <i>z</i><sup>20</sup> = <i>z</i><sup>2</sup>, <i>z</i><sup>25</sup> = <i>z</i> } = <i>G</i>, so that <i>z</i><sup>5</sup> has order 6 and is an alternative generator of <i>G</i>. </p><p>Instead of the <a href="/wiki/Quotient_group" title="Quotient group">quotient</a> notations <b>Z</b>/<i>n</i><b>Z</b>, <b>Z</b>/(<i>n</i>), or <b>Z</b>/<i>n</i>, some authors denote a finite cyclic group as <b>Z</b><sub><i>n</i></sub>, but this clashes with the notation of <a href="/wiki/Number_theory" title="Number theory">number theory</a>, where <b>Z</b><sub><i>p</i></sub> denotes a <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic number</a> ring, or <a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization</a> at a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>. </p> <table class="wikitable" align="left" width="240" style="margin-right: 20px;"> <caption>Infinite cyclic groups </caption> <tbody><tr> <th>p1, (<a href="/wiki/Orbifold_notation" title="Orbifold notation">*∞∞</a>) </th> <th>p11g, (22∞) </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_group_11.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Frieze_group_11.png/120px-Frieze_group_11.png" decoding="async" width="120" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/15/Frieze_group_11.png 1.5x" data-file-width="150" data-file-height="119" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_group_1g.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Frieze_group_1g.png/120px-Frieze_group_1g.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/ee/Frieze_group_1g.png 1.5x" data-file-width="150" data-file-height="120" /></a></span> </td></tr> <tr valign="top"> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_example_p1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Frieze_example_p1.png/120px-Frieze_example_p1.png" decoding="async" width="120" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/10/Frieze_example_p1.png 1.5x" data-file-width="151" data-file-height="49" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:Frieze_hop.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Frieze_hop.png/120px-Frieze_hop.png" decoding="async" width="120" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Frieze_hop.png/180px-Frieze_hop.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Frieze_hop.png/240px-Frieze_hop.png 2x" data-file-width="385" data-file-height="57" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_example_p11g.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Frieze_example_p11g.png/120px-Frieze_example_p11g.png" decoding="async" width="120" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/76/Frieze_example_p11g.png 1.5x" data-file-width="151" data-file-height="49" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:Frieze_step.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Frieze_step.png/120px-Frieze_step.png" decoding="async" width="120" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Frieze_step.png/180px-Frieze_step.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Frieze_step.png/240px-Frieze_step.png 2x" data-file-width="380" data-file-height="117" /></a></span> </td></tr> <tr> <td colspan="2">Two <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a> are isomorphic to <b>Z</b>. With one generator, p1 has translations and p11g has glide reflections. </td></tr></tbody></table> <p>On the other hand, in an <b>infinite cyclic group</b> <span class="nowrap"><i>G</i> = ⟨<i>g</i>⟩</span>, the powers <i>g</i><sup><i>k</i></sup> give distinct elements for all integers <i>k</i>, so that <i>G</i> = { ... , <i>g</i><sup>−2</sup>, <i>g</i><sup>−1</sup>, <i>e</i>, <i>g</i>, <i>g</i><sup>2</sup>, ... }, and <i>G</i> is isomorphic to the standard group <span class="nowrap">C = C<sub>∞</sub></span> and to <b>Z</b>, the additive group of the integers. An example is the first <a href="/wiki/Frieze_group#Descriptions_of_the_seven_frieze_groups" title="Frieze group">frieze group</a>. Here there are no finite cycles, and the name "cyclic" may be misleading.<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> </p><p>To avoid this confusion, <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> introduced the term <b>monogenous group</b> for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".<sup id="cite_ref-algebra1.§4.10_4-0" class="reference"><a href="#cite_note-algebra1.§4.10-4"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right"> <caption>Examples of cyclic groups in rotational symmetry </caption> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Triangle.Scalene.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/120px-Triangle.Scalene.svg.png" decoding="async" width="120" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/180px-Triangle.Scalene.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/240px-Triangle.Scalene.svg.png 2x" data-file-width="245" data-file-height="110" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg/120px-Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg" decoding="async" width="120" height="68" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg/180px-Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg/240px-Hubble2005-01-barred-spiral-galaxy-NGC1300.jpg 2x" data-file-width="6637" data-file-height="3787" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg/120px-The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg.png" decoding="async" width="120" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg/180px-The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg/240px-The_armoured_triskelion_on_the_flag_of_the_Isle_of_Man.svg.png 2x" data-file-width="592" data-file-height="525" /></a></span> </td></tr> <tr> <th><a href="/wiki/Scalene_triangle" class="mw-redirect" title="Scalene triangle">C<sub>1</sub></a> </th> <th><a href="/wiki/NGC_1300" title="NGC 1300">C<sub>2</sub></a> </th> <th><a href="/wiki/Flag_of_the_Isle_of_Man" title="Flag of the Isle of Man">C<sub>3</sub></a> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Circular-cross-decorative-knot-12crossings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circular-cross-decorative-knot-12crossings.svg/120px-Circular-cross-decorative-knot-12crossings.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circular-cross-decorative-knot-12crossings.svg/180px-Circular-cross-decorative-knot-12crossings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circular-cross-decorative-knot-12crossings.svg/240px-Circular-cross-decorative-knot-12crossings.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Flag_of_Hong_Kong.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Flag_of_Hong_Kong.svg/120px-Flag_of_Hong_Kong.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Flag_of_Hong_Kong.svg/180px-Flag_of_Hong_Kong.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Flag_of_Hong_Kong.svg/240px-Flag_of_Hong_Kong.svg.png 2x" data-file-width="900" data-file-height="600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Olavsrose.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Olavsrose.svg/120px-Olavsrose.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Olavsrose.svg/180px-Olavsrose.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Olavsrose.svg/240px-Olavsrose.svg.png 2x" data-file-width="459" data-file-height="459" /></a></span> </td></tr> <tr> <th><a href="/wiki/Celtic_knot" title="Celtic knot">C<sub>4</sub></a> </th> <th><a href="/wiki/Flag_of_Hong_Kong" title="Flag of Hong Kong">C<sub>5</sub></a> </th> <th><a href="https://de.wikipedia.org/wiki/Olavsrose" class="extiw" title="de:Olavsrose">C<sub>6</sub></a> </th></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Integer_and_modular_addition">Integer and modular addition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=3" title="Edit section: Integer and modular addition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of <a href="/wiki/Integer" title="Integer">integers</a> <b>Z</b>, with the operation of addition, forms a group.<sup id="cite_ref-eom_1-2" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is an <b>infinite cyclic group</b>, because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to <b>Z</b>. </p><p>For every positive integer <i>n</i>, the set of integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <i>n</i>, again with the operation of addition, forms a finite cyclic group, denoted <b>Z</b>/<i>n</i><b>Z</b>.<sup id="cite_ref-eom_1-3" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A modular integer <i>i</i> is a generator of this group if <i>i</i> is <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a> to <i>n</i>, because these elements can generate all other elements of the group through integer addition. (The number of such generators is <i>φ</i>(<i>n</i>), where <i>φ</i> is the <a href="/wiki/Euler_totient_function" class="mw-redirect" title="Euler totient function">Euler totient function</a>.) Every finite cyclic group <i>G</i> is isomorphic to <b>Z</b>/<i>n</i><b>Z</b>, where <i>n</i> = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>G</i></span>| is the order of the group. </p><p>The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>, also denoted <b>Z</b> and <b>Z</b>/<i>n</i><b>Z</b> or <b>Z</b>/(<i>n</i>). If <i>p</i> is a <a href="/wiki/Prime_number" title="Prime number">prime</a>, then <b>Z</b>/<i>p<b>Z</b></i> is a <a href="/wiki/Finite_field" title="Finite field">finite field</a>, and is usually denoted <b>F</b><sub><i>p</i></sub> or GF(<i>p</i>) for Galois field. </p> <div class="mw-heading mw-heading3"><h3 id="Modular_multiplication">Modular multiplication</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=4" title="Edit section: Modular multiplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">Multiplicative group of integers modulo n</a></div> <p>For every positive integer <i>n</i>, the set of the integers modulo <i>n</i> that are relatively prime to <i>n</i> is written as (<b>Z</b>/<i>n</i><b>Z</b>)<sup>×</sup>; it <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">forms a group</a> under the operation of multiplication. This group is not always cyclic, but is so whenever <i>n</i> is 1, 2, 4, a <a href="/wiki/Prime_power" title="Prime power">power of an odd prime</a>, or twice a power of an odd prime (sequence <span class="nowrap external"><a href="//oeis.org/A033948" class="extiw" title="oeis:A033948">A033948</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This is the multiplicative group of <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a> of the ring <b>Z</b>/<i>n</i><b>Z</b>; there are <i>φ</i>(<i>n</i>) of them, where again <i>φ</i> is the <a href="/wiki/Euler_totient_function" class="mw-redirect" title="Euler totient function">Euler totient function</a>. For example, (<b>Z</b>/6<b>Z</b>)<sup>×</sup> = {1, 5}, and since 6 is twice an odd prime this is a cyclic group. In contrast, (<b>Z</b>/8<b>Z</b>)<sup>×</sup> = {1, 3, 5, 7} is a <a href="/wiki/Klein_group" class="mw-redirect" title="Klein group">Klein 4-group</a> and is not cyclic. When (<b>Z</b>/<i>n</i><b>Z</b>)<sup>×</sup> is cyclic, its generators are called <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive roots modulo <i>n</i></a>. </p><p>For a prime number <i>p</i>, the group (<b>Z</b>/<i>p</i><b>Z</b>)<sup>×</sup> is always cyclic, consisting of the non-zero elements of the <a href="/wiki/Finite_field" title="Finite field">finite field</a> of order <i>p</i>. More generally, every finite <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the multiplicative group of any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is cyclic.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Rotational_symmetries">Rotational symmetries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=5" title="Edit section: Rotational symmetries"><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">Main article: <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">Rotational symmetry</a></div> <p>The set of <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetries</a> of a <a href="/wiki/Polygon" title="Polygon">polygon</a> forms a finite cyclic group.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> If there are <i>n</i> different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to <b>Z</b>/<i>n</i><b>Z</b>. In three or higher dimensions there exist other <a href="/wiki/Point_groups_in_three_dimensions#Cyclic_3D_symmetry_groups" title="Point groups in three dimensions">finite symmetry groups that are cyclic</a>, but which are not all rotations around an axis, but instead <a href="/wiki/Rotoreflection" class="mw-redirect" title="Rotoreflection">rotoreflections</a>. </p><p>The group of all rotations of a <a href="/wiki/Circle" title="Circle">circle</a> (the <a href="/wiki/Circle_group" title="Circle group">circle group</a>, also denoted <i>S</i><sup>1</sup>) is <i>not</i> cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C<sub>∞</sub> is <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a>, while <i>S</i><sup>1</sup> is not. The group of rotations by rational angles <i>is</i> countable, but still not cyclic. </p> <div class="mw-heading mw-heading3"><h3 id="Galois_theory">Galois theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=6" title="Edit section: Galois theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <i>n</i>th <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a> is a <a href="/wiki/Complex_number" title="Complex number">complex number</a> whose <i>n</i>th power is 1, a <a href="/wiki/Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">root</a> of the <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <span class="nowrap"><i>x</i><sup><i>n</i></sup> − 1</span>. The set of all <i>n</i>th roots of unity forms a cyclic group of order <i>n</i> under multiplication.<sup id="cite_ref-eom_1-4" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The generators of this cyclic group are the <i>n</i>th <a href="/wiki/Primitive_root_of_unity" class="mw-redirect" title="Primitive root of unity">primitive roots of unity</a>; they are the roots of the <i>n</i>th <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomial</a>. For example, the polynomial <span class="nowrap"><i>z</i><sup>3</sup> − 1</span> factors as <span class="nowrap">(<i>z</i> − 1)(<i>z</i> − <i>ω</i>)(<i>z</i> − <i>ω</i><sup>2</sup>)</span>, where <span class="nowrap"><i>ω</i> = <i>e</i><sup>2<i>πi</i>/3</sup></span>; the set {1, <i>ω</i>, <i>ω</i><sup>2</sup>} = {<i>ω</i><sup>0</sup>, <i>ω</i><sup>1</sup>, <i>ω</i><sup>2</sup>} forms a cyclic group under multiplication. The <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of the <a href="/wiki/Field_extension" title="Field extension">field extension</a> of the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> generated by the <i>n</i>th roots of unity forms a different group, isomorphic to the multiplicative group (<b>Z/</b><i>n</i><b>Z</b>)<sup>×</sup> of order <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function"><i>φ</i>(<i>n</i>)</a>, which is cyclic for some but not all <i>n</i> (see above). </p><p>A field extension is called a <a href="/wiki/Cyclic_extension" class="mw-redirect" title="Cyclic extension">cyclic extension</a> if its Galois group is cyclic. For fields of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic zero</a>, such extensions are the subject of <a href="/wiki/Kummer_theory" title="Kummer theory">Kummer theory</a>, and are intimately related to <a href="/wiki/Solvability_by_radicals" class="mw-redirect" title="Solvability by radicals">solvability by radicals</a>. For an extension of <a href="/wiki/Finite_field" title="Finite field">finite fields</a> of characteristic <i>p</i>, its Galois group is always finite and cyclic, generated by a power of the <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius mapping</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Conversely, given a finite field <i>F</i> and a finite cyclic group <i>G</i>, there is a finite field extension of <i>F</i> whose Galois group is <i>G</i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Subgroups">Subgroups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=7" title="Edit section: Subgroups"><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">Main article: <a href="/wiki/Fundamental_theorem_of_cyclic_groups" class="mw-redirect" title="Fundamental theorem of cyclic groups">Fundamental theorem of cyclic groups</a></div> <p>All <a href="/wiki/Subgroup" title="Subgroup">subgroups</a> and <a href="/wiki/Quotient_group" title="Quotient group">quotient groups</a> of cyclic groups are cyclic. Specifically, all subgroups of <b>Z</b> are of the form ⟨<i>m</i>⟩ = <i>m</i><b>Z</b>, with <i>m</i> a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0<b>Z</b>, they all are <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to <b>Z</b>. The <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a> of <b>Z</b> is isomorphic to the <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dual</a> of the lattice of natural numbers ordered by <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Thus, since a prime number <i>p</i> has no nontrivial divisors, <i>p</i><b>Z</b> is a maximal proper subgroup, and the quotient group <b>Z</b>/<i>p</i><b>Z</b> is <a href="/wiki/Simple_group" title="Simple group">simple</a>; in fact, a cyclic group is simple if and only if its order is prime.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>All quotient groups <b>Z</b>/<i>n</i><b>Z</b> are finite, with the exception <span class="nowrap"><b>Z</b>/0<b>Z</b> = <b>Z</b>/{0}.</span> For every positive divisor <i>d</i> of <i>n</i>, the quotient group <b>Z</b>/<i>n</i><b>Z</b> has precisely one subgroup of order <i>d</i>, generated by the <a href="/wiki/Modular_arithmetic#Congruence_classes" title="Modular arithmetic">residue class</a> of <i>n</i>/<i>d</i>. There are no other subgroups. </p> <div class="mw-heading mw-heading2"><h2 id="Additional_properties">Additional properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=8" title="Edit section: Additional properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every cyclic group is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>.<sup id="cite_ref-eom_1-5" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> That is, its group operation is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>: <span class="nowrap"><i>gh</i> = <i>hg</i></span> (for all <i>g</i> and <i>h</i> in <i>G</i>). This is clear for the groups of integer and modular addition since <span class="nowrap"><i>r</i> + <i>s</i> ≡ <i>s</i> + <i>r</i> (mod <i>n</i>)</span>, and it follows for all cyclic groups since they are all isomorphic to these standard groups. For a finite cyclic group of order <i>n</i>, <i>g</i><sup><i>n</i></sup> is the identity element for any element <i>g</i>. This again follows by using the isomorphism to modular addition, since <span class="nowrap"><i>kn</i> ≡ 0 (mod <i>n</i>)</span> for every integer <i>k</i>. (This is also true for a general group of order <i>n</i>, due to <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a>.) </p><p>For a <a href="/wiki/Prime_power" title="Prime power">prime power</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 p^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e017c102135ab13bdf501dc1c1b5fd1840a97822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:3.009ex;" alt="{\displaystyle p^{k}}"></span>, the 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 Z/p^{k}Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z/p^{k}Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec066a6769ab59505df90e2a0fee0527076acc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.781ex; height:3.176ex;" alt="{\displaystyle Z/p^{k}Z}"></span> is called a <a href="/wiki/Primary_cyclic_group" title="Primary cyclic group">primary cyclic group</a>. The <a href="/wiki/Fundamental_theorem_of_finitely_generated_abelian_groups" class="mw-redirect" title="Fundamental theorem of finitely generated abelian groups">fundamental theorem of abelian groups</a> states that every <a href="/wiki/Finitely_generated_abelian_group" title="Finitely generated abelian group">finitely generated abelian group</a> is a finite direct product of primary cyclic and infinite cyclic groups. </p><p>Because a cyclic group is abelian, each of its <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy classes</a> consists of a single element. A cyclic group of order <i>n</i> therefore has <i>n</i> conjugacy classes. </p><p>If <i>d</i> is a <a href="/wiki/Divisor" title="Divisor">divisor</a> of <i>n</i>, then the number of elements in <b>Z</b>/<i>n</i><b>Z</b> which have order <i>d</i> is <i>φ</i>(<i>d</i>), and the number of elements whose order divides <i>d</i> is exactly <i>d</i>. If <i>G</i> is a finite group in which, for each <span class="nowrap"><i>n</i> > 0</span>, <i>G</i> contains at most <i>n</i> elements of order dividing <i>n</i>, then <i>G</i> must be cyclic.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> The order of an element <i>m</i> in <b>Z</b>/<i>n</i><b>Z</b> is <i>n</i>/<a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>(<i>n</i>,<i>m</i>). </p><p>If <i>n</i> and <i>m</i> are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>, then the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of two cyclic groups <b>Z</b>/<i>n</i><b>Z</b> and <b>Z</b>/<i>m</i><b>Z</b> is isomorphic to the cyclic group <b>Z</b>/<i>nm</i><b>Z</b>, and the converse also holds: this is one form of the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>. For example, <b>Z</b>/12<b>Z</b> is isomorphic to the direct product <span class="nowrap"><b>Z</b>/3<b>Z</b> × <b>Z</b>/4<b>Z</b></span> under the isomorphism <span class="nowrap">(<i>k</i> mod 12) → (<i>k</i> mod 3, <i>k</i> mod 4)</span>; but it is not isomorphic to <span class="nowrap"><b>Z</b>/6<b>Z</b> × <b>Z</b>/2<b>Z</b></span>, in which every element has order at most 6. </p><p>If <i>p</i> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a>, then any group with <i>p</i> elements is isomorphic to the simple group <b>Z</b>/<i>p</i><b>Z</b>. A number <i>n</i> is called a <a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">cyclic number</a> if <b>Z</b>/<i>n</i><b>Z</b> is the only group of order <i>n</i>, which is true exactly when <span class="nowrap">gcd(<i>n</i>, <i>φ</i>(<i>n</i>)) = 1</span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The sequence of cyclic numbers include all primes, but some are <a href="/wiki/Composite_number" title="Composite number">composite</a> such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are: </p> <dl><dd>1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... (sequence <span class="nowrap external"><a href="//oeis.org/A003277" class="extiw" title="oeis:A003277">A003277</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>The definition immediately implies that cyclic groups have <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">group presentation</a> <span class="nowrap">C<sub>∞</sub> = ⟨<i>x</i> | ⟩</span> and <span class="nowrap">C<sub><i>n</i></sub> = ⟨<i>x</i> | <i>x</i><sup><i>n</i></sup>⟩</span> for finite <i>n</i>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Associated_objects">Associated objects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=9" title="Edit section: Associated objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Representations">Representations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=10" title="Edit section: Representations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Representation_theory_of_finite_groups" title="Representation theory of finite groups">representation theory</a> of the cyclic group is a critical base case for the representation theory of more general finite groups. In the <a href="/wiki/Character_theory" title="Character theory">complex case</a>, a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent. In the <a href="/wiki/Modular_representation_theory" title="Modular representation theory">positive characteristic case</a>, the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic <a href="/wiki/Sylow_subgroup" class="mw-redirect" title="Sylow subgroup">Sylow subgroups</a> and more generally the representation theory of blocks of cyclic defect. </p> <div class="mw-heading mw-heading3"><h3 id="Cycle_graph">Cycle graph</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=11" title="Edit section: Cycle graph"><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/Cycle_graph_(algebra)" title="Cycle graph (algebra)">Cycle graph (algebra)</a></div> <p><span class="anchor" id="Z_sub2"></span> A <b>cycle graph</b> illustrates the various cycles of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and is particularly useful in visualizing the structure of small <a href="/wiki/Finite_group" title="Finite group">finite groups</a>. A cycle graph for a cyclic group is simply a <a href="/wiki/Circular_graph" class="mw-redirect" title="Circular graph">circular graph</a>, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse generator defines a backwards path. A trivial path (identity) can be drawn as a <a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">loop</a> but is usually suppressed. Z<sub>2</sub> is sometimes drawn with two curved edges as a <a href="/wiki/Multigraph" title="Multigraph">multigraph</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>A cyclic group Z<sub><i>n</i></sub>, with order <i>n</i>, corresponds to a single cycle graphed simply as an <i>n</i>-sided polygon with the elements at the vertices. </p> <table class="wikitable"> <caption>Cycle graphs up to order 24 </caption> <tbody><tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/40px-GroupDiagramMiniC1.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/60px-GroupDiagramMiniC1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/80px-GroupDiagramMiniC1.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <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/60px-GroupDiagramMiniC2.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/90px-GroupDiagramMiniC2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/120px-GroupDiagramMiniC2.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/60px-GroupDiagramMiniC3.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/90px-GroupDiagramMiniC3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/120px-GroupDiagramMiniC3.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/GroupDiagramMiniC4.svg/60px-GroupDiagramMiniC4.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/GroupDiagramMiniC4.svg/90px-GroupDiagramMiniC4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/GroupDiagramMiniC4.svg/120px-GroupDiagramMiniC4.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/60px-GroupDiagramMiniC5.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/90px-GroupDiagramMiniC5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/120px-GroupDiagramMiniC5.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/60px-GroupDiagramMiniC6.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/90px-GroupDiagramMiniC6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/120px-GroupDiagramMiniC6.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC7.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/GroupDiagramMiniC7.svg/60px-GroupDiagramMiniC7.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/GroupDiagramMiniC7.svg/90px-GroupDiagramMiniC7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/GroupDiagramMiniC7.svg/120px-GroupDiagramMiniC7.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/GroupDiagramMiniC8.svg/60px-GroupDiagramMiniC8.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/GroupDiagramMiniC8.svg/90px-GroupDiagramMiniC8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/GroupDiagramMiniC8.svg/120px-GroupDiagramMiniC8.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td></tr> <tr align="center"> <td>Z<sub>1</sub></td> <td>Z<sub>2</sub></td> <td>Z<sub>3</sub></td> <td>Z<sub>4</sub></td> <td>Z<sub>5</sub></td> <td>Z<sub>6</sub> = Z<sub>3</sub>×Z<sub>2</sub></td> <td>Z<sub>7</sub></td> <td>Z<sub>8</sub> </td></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC9.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GroupDiagramMiniC9.svg/60px-GroupDiagramMiniC9.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GroupDiagramMiniC9.svg/90px-GroupDiagramMiniC9.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GroupDiagramMiniC9.svg/120px-GroupDiagramMiniC9.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/60px-GroupDiagramMiniC10.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/90px-GroupDiagramMiniC10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/120px-GroupDiagramMiniC10.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC11.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/GroupDiagramMiniC11.svg/60px-GroupDiagramMiniC11.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/GroupDiagramMiniC11.svg/90px-GroupDiagramMiniC11.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/GroupDiagramMiniC11.svg/120px-GroupDiagramMiniC11.svg.png 2x" data-file-width="32" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniC12.svg/60px-GroupDiagramMiniC12.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniC12.svg/90px-GroupDiagramMiniC12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniC12.svg/120px-GroupDiagramMiniC12.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC13.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/GroupDiagramMiniC13.svg/60px-GroupDiagramMiniC13.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/GroupDiagramMiniC13.svg/90px-GroupDiagramMiniC13.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/GroupDiagramMiniC13.svg/120px-GroupDiagramMiniC13.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC14.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/GroupDiagramMiniC14.svg/60px-GroupDiagramMiniC14.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/GroupDiagramMiniC14.svg/90px-GroupDiagramMiniC14.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/GroupDiagramMiniC14.svg/120px-GroupDiagramMiniC14.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC15.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC15.svg/60px-GroupDiagramMiniC15.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC15.svg/90px-GroupDiagramMiniC15.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC15.svg/120px-GroupDiagramMiniC15.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/GroupDiagramMiniC16.svg/60px-GroupDiagramMiniC16.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/GroupDiagramMiniC16.svg/90px-GroupDiagramMiniC16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/GroupDiagramMiniC16.svg/120px-GroupDiagramMiniC16.svg.png 2x" data-file-width="64" data-file-height="64" /></a></span> </td></tr> <tr align="center"> <td>Z<sub>9</sub></td> <td>Z<sub>10</sub> = Z<sub>5</sub>×Z<sub>2</sub></td> <td>Z<sub>11</sub></td> <td>Z<sub>12</sub> = Z<sub>4</sub>×Z<sub>3</sub></td> <td>Z<sub>13</sub></td> <td>Z<sub>14</sub> = Z<sub>7</sub>×Z<sub>2</sub></td> <td>Z<sub>15</sub> = Z<sub>5</sub>×Z<sub>3</sub></td> <td>Z<sub>16</sub> </td></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC17.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GroupDiagramMiniC17.svg/60px-GroupDiagramMiniC17.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GroupDiagramMiniC17.svg/90px-GroupDiagramMiniC17.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GroupDiagramMiniC17.svg/120px-GroupDiagramMiniC17.svg.png 2x" data-file-width="996" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC18.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/GroupDiagramMiniC18.svg/60px-GroupDiagramMiniC18.svg.png" decoding="async" width="60" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/GroupDiagramMiniC18.svg/90px-GroupDiagramMiniC18.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/GroupDiagramMiniC18.svg/120px-GroupDiagramMiniC18.svg.png 2x" data-file-width="1000" data-file-height="985" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC19.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/GroupDiagramMiniC19.svg/60px-GroupDiagramMiniC19.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/GroupDiagramMiniC19.svg/90px-GroupDiagramMiniC19.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/GroupDiagramMiniC19.svg/120px-GroupDiagramMiniC19.svg.png 2x" data-file-width="997" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC20.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC20.svg/60px-GroupDiagramMiniC20.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC20.svg/90px-GroupDiagramMiniC20.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC20.svg/120px-GroupDiagramMiniC20.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC21.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/GroupDiagramMiniC21.svg/60px-GroupDiagramMiniC21.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/GroupDiagramMiniC21.svg/90px-GroupDiagramMiniC21.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/GroupDiagramMiniC21.svg/120px-GroupDiagramMiniC21.svg.png 2x" data-file-width="997" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC22.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/GroupDiagramMiniC22.svg/60px-GroupDiagramMiniC22.svg.png" decoding="async" width="60" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/GroupDiagramMiniC22.svg/90px-GroupDiagramMiniC22.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/GroupDiagramMiniC22.svg/120px-GroupDiagramMiniC22.svg.png 2x" data-file-width="1000" data-file-height="990" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC23.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/GroupDiagramMiniC23.svg/60px-GroupDiagramMiniC23.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/GroupDiagramMiniC23.svg/90px-GroupDiagramMiniC23.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/GroupDiagramMiniC23.svg/120px-GroupDiagramMiniC23.svg.png 2x" data-file-width="998" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC24.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/GroupDiagramMiniC24.svg/60px-GroupDiagramMiniC24.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/GroupDiagramMiniC24.svg/90px-GroupDiagramMiniC24.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/GroupDiagramMiniC24.svg/120px-GroupDiagramMiniC24.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td></tr> <tr align="center"> <td>Z<sub>17</sub></td> <td>Z<sub>18</sub> = Z<sub>9</sub>×Z<sub>2</sub></td> <td>Z<sub>19</sub></td> <td>Z<sub>20</sub> = Z<sub>5</sub>×Z<sub>4</sub></td> <td>Z<sub>21</sub> = Z<sub>7</sub>×Z<sub>3</sub></td> <td>Z<sub>22</sub> = Z<sub>11</sub>×Z<sub>2</sub></td> <td>Z<sub>23</sub></td> <td>Z<sub>24</sub> = Z<sub>8</sub>×Z<sub>3</sub> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Cayley_graph">Cayley graph</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=12" title="Edit section: Cayley graph"><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">Main article: <a href="/wiki/Circulant_graph" title="Circulant graph">Circulant graph</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Paley13.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Paley13.svg/240px-Paley13.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Paley13.svg/360px-Paley13.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Paley13.svg/480px-Paley13.svg.png 2x" data-file-width="495" data-file-height="495" /></a><figcaption>The <a href="/wiki/Paley_graph" title="Paley graph">Paley graph</a> of order 13, a circulant graph formed as the Cayley graph of <b>Z</b>/13 with generator set {1,3,4}</figcaption></figure> <p>A <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a> is a graph defined from a pair (<i>G</i>,<i>S</i>) where <i>G</i> is a group and <i>S</i> is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a <a href="/wiki/Cycle_graph" title="Cycle graph">cycle graph</a>, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite <a href="/wiki/Path_graph" title="Path graph">path graph</a>. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called <a href="/wiki/Circulant_graph" title="Circulant graph">circulant graphs</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the <a href="/wiki/Vertex-transitive_graph" title="Vertex-transitive graph">vertex-transitive graphs</a> whose <a href="/wiki/Graph_automorphism" title="Graph automorphism">symmetry group</a> includes a transitive cyclic group.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Endomorphisms">Endomorphisms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=13" title="Edit section: Endomorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of the abelian group <b>Z</b>/<i>n</i><b>Z</b> is <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">isomorphic</a> to <b>Z</b>/<i>n</i><b>Z</b> itself as a <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring</a>.<sup id="cite_ref-ks_20-0" class="reference"><a href="#cite_note-ks-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Under this isomorphism, the number <i>r</i> corresponds to the endomorphism of <b>Z</b>/<i>n</i><b>Z</b> that maps each element to the sum of <i>r</i> copies of it. This is a <a href="/wiki/Bijection" title="Bijection">bijection</a> if and only if <i>r</i> is coprime with <i>n</i>, so the <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of <b>Z</b>/<i>n</i><b>Z</b> is isomorphic to the unit group (<b>Z</b>/<i>n</i><b>Z</b>)<sup>×</sup>.<sup id="cite_ref-ks_20-1" class="reference"><a href="#cite_note-ks-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Similarly, the endomorphism ring of the additive group of <b>Z</b> is isomorphic to the ring <b>Z</b>. Its automorphism group is isomorphic to the group of units of the ring <b>Z</b>, which is <span class="nowrap">({−1, +1}, ×) ≅ C<sub>2</sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Tensor_product_and_Hom_of_cyclic_groups">Tensor product and Hom of cyclic groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=14" title="Edit section: Tensor product and Hom of cyclic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> <span class="nowrap"><b>Z</b>/<i>m</i><b>Z</b> ⊗ <b>Z</b>/<i>n</i><b>Z</b></span> can be shown to be isomorphic to <span class="nowrap"><b>Z</b> / gcd(<i>m</i>, <i>n</i>)<b>Z</b></span>. So we can form the collection of group <a href="/wiki/Group_homomorphism" title="Group homomorphism">homomorphisms</a> from <b>Z</b>/<i>m</i><b>Z</b> to <b>Z</b>/<i>n</i><b>Z</b>, denoted <span class="nowrap">hom(<b>Z</b>/<i>m</i><b>Z</b>, <b>Z</b>/<i>n</i><b>Z</b>)</span>, which is itself a group. </p><p>For the tensor product, this is a consequence of the general fact that <span class="nowrap"><i>R</i>/<i>I</i> ⊗<sub><i>R</i></sub> <i>R</i>/<i>J</i> ≅ <i>R</i>/(<i>I</i> + <i>J</i>)</span>, where <i>R</i> is a commutative <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> with unit and <i>I</i> and <i>J</i> are <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> of the ring. For the Hom group, recall that it is isomorphic to the subgroup of <span class="nowrap"><b>Z</b> / <i>n</i><b>Z</b></span> consisting of the elements of order dividing <i>m</i>. That subgroup is cyclic of order <span class="nowrap">gcd(<i>m</i>, <i>n</i>)</span>, which completes the proof. </p> <div class="mw-heading mw-heading2"><h2 id="Related_classes_of_groups">Related classes of groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=15" title="Edit section: Related classes of groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several other classes of groups have been defined by their relation to the cyclic groups: </p> <div class="mw-heading mw-heading3"><h3 id="Virtually_cyclic_groups">Virtually cyclic groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=16" title="Edit section: Virtually cyclic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A group is called <b>virtually cyclic</b> if it contains a cyclic subgroup of finite <a href="/wiki/Index_(group_theory)" class="mw-redirect" title="Index (group theory)">index</a> (the number of <a href="/wiki/Coset" title="Coset">cosets</a> that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by multiplying a member of the cyclic subgroup and a member of a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a> and has exactly two <a href="/wiki/End_(graph_theory)" title="End (graph theory)">ends</a>;<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> an example of such a group is the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of <b>Z</b>/<i>n</i><b>Z</b> and <b>Z</b>, in which the factor <b>Z</b> has finite index <i>n</i>. Every abelian subgroup of a <a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Gromov hyperbolic group</a> is virtually cyclic.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Procyclic_groups">Procyclic groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=17" title="Edit section: Procyclic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Profinite_group" title="Profinite group">profinite group</a> is called <b>procyclic</b> if it can be topologically generated by a single element. Examples of profinite groups include the profinite integers <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 {\widehat {\mathbb {Z} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12112fb1afd12580b51b080bb33356edd1d0c797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }}}"></span> or the <i>p</i>-adic integers <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} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></span> for a prime number <i>p</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Locally_cyclic_groups">Locally cyclic groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=18" title="Edit section: Locally cyclic groups"><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">Main article: <a href="/wiki/Locally_cyclic_group" title="Locally cyclic group">Locally cyclic group</a></div> <p>A <a href="/wiki/Locally_cyclic_group" title="Locally cyclic group">locally cyclic group</a> is a group in which each <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a> subgroup is cyclic. An example is the additive group of the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>: every finite set of rational numbers is a set of integer multiples of a single <a href="/wiki/Unit_fraction" title="Unit fraction">unit fraction</a>, the inverse of their <a href="/wiki/Lowest_common_denominator" title="Lowest common denominator">lowest common denominator</a>, and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a> is a <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Cyclically_ordered_groups">Cyclically ordered groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=19" title="Edit section: Cyclically ordered groups"><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">Main article: <a href="/wiki/Cyclically_ordered_group" title="Cyclically ordered group">Cyclically ordered group</a></div> <p>A <a href="/wiki/Cyclically_ordered_group" title="Cyclically ordered group">cyclically ordered group</a> is a group together with a <a href="/wiki/Cyclic_order" title="Cyclic order">cyclic order</a> preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group). Every finite subgroup of a cyclically ordered group is cyclic.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Metacyclic_and_polycyclic_groups">Metacyclic and polycyclic groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=20" title="Edit section: Metacyclic and polycyclic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Metacyclic_group" title="Metacyclic group">metacyclic group</a> is a group containing a cyclic <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> whose quotient is also cyclic.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> These groups include the cyclic groups, the <a href="/wiki/Dicyclic_group" title="Dicyclic group">dicyclic groups</a>, and the <a href="/wiki/Direct_product" title="Direct product">direct products</a> of two cyclic groups. The <a href="/wiki/Polycyclic_group" title="Polycyclic group">polycyclic groups</a> generalize metacyclic groups by allowing more than one level of <a href="/wiki/Group_extension" title="Group extension">group extension</a>. A group is polycyclic if it has a finite descending sequence of subgroups, each of which is normal in the previous subgroup with a cyclic quotient, ending in the trivial group. Every finitely generated <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> or <a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent group</a> is polycyclic.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </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=Cyclic_group&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cycle_graph_(group)" class="mw-redirect" title="Cycle graph (group)">Cycle graph (group)</a></li> <li><a href="/wiki/Cyclic_module" title="Cyclic module">Cyclic module</a></li> <li><a href="/wiki/Cyclic_sieving" title="Cyclic sieving">Cyclic sieving</a></li> <li><a href="/wiki/Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a> (<a href="/wiki/Countable_set" title="Countable set">countably infinite</a> analogue)</li> <li><a href="/wiki/Circle_group" title="Circle group">Circle group</a> (<a href="/wiki/Uncountable_set" title="Uncountable set">uncountably infinite</a> analogue)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=22" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=23" title="Edit section: Notes"><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"><ol class="references"> <li id="cite_note-algebra1.§4.10-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-algebra1.§4.10_4-0">^</a></b></span> <span class="reference-text">DEFINITION 15. <i>A group is called</i> monogenous <i>if it admits a system of generators consisting of a single element. A finite monogenous group is called</i> cyclic.<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></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">This implication remains true even if only prime values of <i>n</i> are considered.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> (And observe that when <i>n</i> is prime, there is exactly one element whose order is a proper divisor of <i>n</i>, namely the identity.)</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">If <i>G</i> has two ends, the explicit structure of <i>G</i> is well known: <i>G</i> is an extension of a finite group by either the infinite cyclic group or the infinite dihedral group.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_group&action=edit&section=24" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-eom-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-eom_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-eom_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-eom_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-eom_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-eom_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-eom_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><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 class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Cyclic_group">"Cyclic group"</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=Cyclic+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DCyclic_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></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">(<a href="#CITEREFLajoieMura2000">Lajoie & Mura 2000</a>, pp. 29–33).</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">(<a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, p. 49) or <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=STS9aZ6F204C&pg=PA49">Algebra I: Chapters 1–3</a></i>, p. 49, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a>.</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">(<a href="#CITEREFMotwaniRaghavan1995">Motwani & Raghavan 1995</a>, p. 401).</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">(<a href="#CITEREFVinogradov2003">Vinogradov 2003</a>, pp. 105–132, § VI PRIMITIVE ROOTS AND INDICES).</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="#CITEREFRotman1998">Rotman 1998</a>, p. 65).</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">(<a href="#CITEREFStewartGolubitsky2010">Stewart & Golubitsky 2010</a>, pp. 47–48).</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="#CITEREFCox2012">Cox 2012</a>, p. 294, Theorem 11.1.7).</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">(<a href="#CITEREFCox2012">Cox 2012</a>, p. 295, Corollary 11.1.8 and Theorem 11.1.9).</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">(<a href="#CITEREFAluffi2009">Aluffi 2009</a>, pp. 82–84, 6.4 Example: Subgroups of Cyclic Groups).</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">(<a href="#CITEREFGannon2006">Gannon 2006</a>, p. 18).</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">(<a href="#CITEREFGallian2010">Gallian 2010</a>, p. 84, Exercise 43).</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">(<a href="#CITEREFJungnickel1992">Jungnickel 1992</a>, pp. 545–547).</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">(<a href="#CITEREFCoxeterMoser1980">Coxeter & Moser 1980</a>, p. 1).</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Cycle_Graph"><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/CycleGraph.html">"Cycle Graph"</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=Cycle+Graph&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCycleGraph.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">(<a href="#CITEREFAlspach1997">Alspach 1997</a>, pp. 1–22).</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">(<a href="#CITEREFVilfred2004">Vilfred 2004</a>, pp. 34–36).</span> </li> <li id="cite_note-ks-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-ks_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ks_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(<a href="#CITEREFKurzweilStellmacher2004">Kurzweil & Stellmacher 2004</a>, p. 50).</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">(<a href="#CITEREFStallings1970">Stallings 1970</a>, pp. 124–128). See in particular <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=3Lyvsc694T4C&pg=PA126">Groups of cohomological dimension one</a></i>, p. 126, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a>.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">(<a href="#CITEREFAlonso1991">Alonso 1991</a>, Corollary 3.6).</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">(<a href="#CITEREFOre1938">Ore 1938</a>, pp. 247–269).</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">(<a href="#CITEREFFuchs2011">Fuchs 2011</a>, p. 63).</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFA._L._Shmel'kin2001" class="citation cs2">A. L. Shmel'kin (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Metacyclic_group">"Metacyclic group"</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></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Metacyclic+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.au=A.+L.+Shmel%27kin&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMetacyclic_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><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=Polycyclic_group">"Polycyclic group"</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=Polycyclic+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPolycyclic_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></span> </li> </ol></div> <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=Cyclic_group&action=edit&section=25" title="Edit section: References"><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="CITEREFAlonso1991" class="citation cs2">Alonso, J. M.; et al. (1991), "Notes on word hyperbolic groups", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130425163632/http://www.cmi.univ-mrs.fr/~hamish/Papers/MSRInotes2004.pdf"><i>Group theory from a geometrical viewpoint (Trieste, 1990)</i></a> <span class="cs1-format">(PDF)</span>, River Edge, NJ: World Scientific, Corollary 3.6, <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=1170363">1170363</a>, archived from <a rel="nofollow" class="external text" href="http://www.cmi.univ-mrs.fr/~hamish/Papers/MSRInotes2004.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-04-25<span class="reference-accessdate">, retrieved <span class="nowrap">2013-11-26</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Notes+on+word+hyperbolic+groups&rft.btitle=Group+theory+from+a+geometrical+viewpoint+%28Trieste%2C+1990%29&rft.place=River+Edge%2C+NJ&rft.pages=Corollary-3.6&rft.pub=World+Scientific&rft.date=1991&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1170363%23id-name%3DMR&rft.aulast=Alonso&rft.aufirst=J.+M.&rft_id=http%3A%2F%2Fwww.cmi.univ-mrs.fr%2F~hamish%2FPapers%2FMSRInotes2004.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlspach1997" class="citation cs2"><a href="/wiki/Brian_Alspach" title="Brian Alspach">Alspach, Brian</a> (1997), "Isomorphism and Cayley graphs on abelian groups", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-tIaXdII8egC&pg=PA1"><i>Graph symmetry (Montreal, PQ, 1996)</i></a>, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 497, Dordrecht: Kluwer Acad. Publ., pp. 1–22, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-792-34668-5" title="Special:BookSources/978-0-792-34668-5"><bdi>978-0-792-34668-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1468786">1468786</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Isomorphism+and+Cayley+graphs+on+abelian+groups&rft.btitle=Graph+symmetry+%28Montreal%2C+PQ%2C+1996%29&rft.place=Dordrecht&rft.series=NATO+Adv.+Sci.+Inst.+Ser.+C+Math.+Phys.+Sci.&rft.pages=1-22&rft.pub=Kluwer+Acad.+Publ.&rft.date=1997&rft.isbn=978-0-792-34668-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1468786%23id-name%3DMR&rft.aulast=Alspach&rft.aufirst=Brian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-tIaXdII8egC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAluffi2009" class="citation cs2">Aluffi, Paolo (2009), "6.4 Example: Subgroups of Cyclic Groups", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=deWkZWYbyHQC&pg=PA82"><i>Algebra, Chapter 0</i></a>, <a href="/wiki/Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 104, American Mathematical Society, pp. 82–84, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4781-7" title="Special:BookSources/978-0-8218-4781-7"><bdi>978-0-8218-4781-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6.4+Example%3A+Subgroups+of+Cyclic+Groups&rft.btitle=Algebra%2C+Chapter+0&rft.series=Graduate+Studies+in+Mathematics&rft.pages=82-84&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4781-7&rft.aulast=Aluffi&rft.aufirst=Paolo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdeWkZWYbyHQC%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1998" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1998-08-03) [1970], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=STS9aZ6F204C&pg=PA49"><i>Algebra I: Chapters 1-3</i></a>, Elements of Mathematics, vol. 1 (softcover reprint ed.), Springer Science & Business Media, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-64243-5" title="Special:BookSources/978-3-540-64243-5"><bdi>978-3-540-64243-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+I%3A+Chapters+1-3&rft.series=Elements+of+Mathematics&rft.edition=softcover+reprint&rft.pub=Springer+Science+%26+Business+Media&rft.date=1998-08-03&rft.isbn=978-3-540-64243-5&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSTS9aZ6F204C%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeterMoser1980" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a>; Moser, W. O. J. (1980), <i>Generators and Relations for Discrete Groups</i>, New York: Springer-Verlag, p. 1, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-09212-9" title="Special:BookSources/0-387-09212-9"><bdi>0-387-09212-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generators+and+Relations+for+Discrete+Groups&rft.place=New+York&rft.pages=1&rft.pub=Springer-Verlag&rft.date=1980&rft.isbn=0-387-09212-9&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rft.au=Moser%2C+W.+O.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLajoieMura2000" class="citation cs2">Lajoie, Caroline; Mura, Roberta (November 2000), "What's in a name? A learning difficulty in connection with cyclic groups", <i>For the Learning of Mathematics</i>, <b>20</b> (3): 29–33, <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/40248334">40248334</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=For+the+Learning+of+Mathematics&rft.atitle=What%27s+in+a+name%3F+A+learning+difficulty+in+connection+with+cyclic+groups&rft.volume=20&rft.issue=3&rft.pages=29-33&rft.date=2000-11&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F40248334%23id-name%3DJSTOR&rft.aulast=Lajoie&rft.aufirst=Caroline&rft.au=Mura%2C+Roberta&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCox2012" class="citation cs2"><a href="/wiki/David_A._Cox" title="David A. Cox">Cox, David A.</a> (2012), <i>Galois Theory</i>, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, Theorem 11.1.7, p. 294, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781118218457">10.1002/9781118218457</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-07205-9" title="Special:BookSources/978-1-118-07205-9"><bdi>978-1-118-07205-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory&rft.series=Pure+and+Applied+Mathematics&rft.pages=Theorem-11.1.7%2C+p.-294&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=2012&rft_id=info%3Adoi%2F10.1002%2F9781118218457&rft.isbn=978-1-118-07205-9&rft.aulast=Cox&rft.aufirst=David+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallian2010" class="citation cs2">Gallian, Joseph (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CnH3mlOKpsMC&pg=PA84"><i>Contemporary Abstract Algebra</i></a> (7th ed.), Cengage Learning, Exercise 43, p. 84, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16509-7" title="Special:BookSources/978-0-547-16509-7"><bdi>978-0-547-16509-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Abstract+Algebra&rft.pages=Exercise-43%2C+p.-84&rft.edition=7th&rft.pub=Cengage+Learning&rft.date=2010&rft.isbn=978-0-547-16509-7&rft.aulast=Gallian&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCnH3mlOKpsMC%26pg%3DPA84&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGannon2006" class="citation cs2">Gannon, Terry (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ehrUt21SnsoC&pg=PA18"><i>Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics</i></a>, Cambridge monographs on mathematical physics, Cambridge University Press, p. 18, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83531-2" title="Special:BookSources/978-0-521-83531-2"><bdi>978-0-521-83531-2</bdi></a>, <q><b>Z</b><sub><i>n</i></sub> is simple iff <i>n</i> is prime.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Moonshine+beyond+the+monster%3A+the+bridge+connecting+algebra%2C+modular+forms+and+physics&rft.series=Cambridge+monographs+on+mathematical+physics&rft.pages=18&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-83531-2&rft.aulast=Gannon&rft.aufirst=Terry&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DehrUt21SnsoC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJungnickel1992" class="citation cs2"><a href="/wiki/Dieter_Jungnickel" title="Dieter Jungnickel">Jungnickel, Dieter</a> (1992), "On the uniqueness of the cyclic group of order <i>n</i>", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>99</b> (6): 545–547, <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%2F2324062">10.2307/2324062</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/2324062">2324062</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=1166004">1166004</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=On+the+uniqueness+of+the+cyclic+group+of+order+n&rft.volume=99&rft.issue=6&rft.pages=545-547&rft.date=1992&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1166004%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2324062%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2324062&rft.aulast=Jungnickel&rft.aufirst=Dieter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuchs2011" class="citation cs2"><a href="/wiki/L%C3%A1szl%C3%B3_Fuchs" title="László Fuchs">Fuchs, László</a> (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V_k79sVPcqYC&pg=PA63"><i>Partially Ordered Algebraic Systems</i></a>, International series of monographs in pure and applied mathematics, vol. 28, Courier Dover Publications, p. 63, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-48387-0" title="Special:BookSources/978-0-486-48387-0"><bdi>978-0-486-48387-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partially+Ordered+Algebraic+Systems&rft.series=International+series+of+monographs+in+pure+and+applied+mathematics&rft.pages=63&rft.pub=Courier+Dover+Publications&rft.date=2011&rft.isbn=978-0-486-48387-0&rft.aulast=Fuchs&rft.aufirst=L%C3%A1szl%C3%B3&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DV_k79sVPcqYC%26pg%3DPA63&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurzweilStellmacher2004" class="citation cs2">Kurzweil, Hans; Stellmacher, Bernd (2004), 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id="CITEREFMotwaniRaghavan1995" class="citation cs2"><a href="/wiki/Rajeev_Motwani" title="Rajeev Motwani">Motwani, Rajeev</a>; Raghavan, Prabhakar (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QKVY4mDivBEC&pg=PA401"><i>Randomized Algorithms</i></a>, Cambridge University Press, Theorem 14.14, p. 401, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-47465-8" title="Special:BookSources/978-0-521-47465-8"><bdi>978-0-521-47465-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Randomized+Algorithms&rft.pages=Theorem-14.14%2C+p.-401&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=978-0-521-47465-8&rft.aulast=Motwani&rft.aufirst=Rajeev&rft.au=Raghavan%2C+Prabhakar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQKVY4mDivBEC%26pg%3DPA401&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOre1938" class="citation cs2"><a href="/wiki/%C3%98ystein_Ore" title="Øystein Ore">Ore, Øystein</a> (1938), "Structures and group theory. II", <i>Duke Mathematical Journal</i>, <b>4</b> (2): 247–269, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-38-00419-3">10.1215/S0012-7094-38-00419-3</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F100155">10338.dmlcz/100155</a></span>, <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=1546048">1546048</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=Structures+and+group+theory.+II&rft.volume=4&rft.issue=2&rft.pages=247-269&rft.date=1938&rft_id=info%3Ahdl%2F10338.dmlcz%2F100155&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1546048%23id-name%3DMR&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-38-00419-3&rft.aulast=Ore&rft.aufirst=%C3%98ystein&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman1998" class="citation cs2">Rotman, Joseph J. (1998), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M32GNlFkmHgC&pg=PA65"><i>Galois Theory</i></a>, Universitext, Springer, Theorem 62, p. 65, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98541-1" title="Special:BookSources/978-0-387-98541-1"><bdi>978-0-387-98541-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory&rft.series=Universitext&rft.pages=Theorem-62%2C+p.-65&rft.pub=Springer&rft.date=1998&rft.isbn=978-0-387-98541-1&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DM32GNlFkmHgC%26pg%3DPA65&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStallings1970" class="citation cs2">Stallings, John (1970), "Groups of cohomological dimension one", <i>Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVIII, New York, 1968)</i>, Providence, R.I.: Amer. Math. Soc., pp. 124–128, <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=0255689">0255689</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Groups+of+cohomological+dimension+one&rft.btitle=Applications+of+Categorical+Algebra+%28Proc.+Sympos.+Pure+Math.%2C+Vol.+XVIII%2C+New+York%2C+1968%29&rft.place=Providence%2C+R.I.&rft.pages=124-128&rft.pub=Amer.+Math.+Soc.&rft.date=1970&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0255689%23id-name%3DMR&rft.aulast=Stallings&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewartGolubitsky2010" class="citation cs2"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a>; <a href="/wiki/Marty_Golubitsky" title="Marty Golubitsky">Golubitsky, Martin</a> (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7x_MF83tTKQC&pg=PA47"><i>Fearful Symmetry: Is God a Geometer?</i></a>, Courier Dover Publications, pp. 47–48, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47758-9" title="Special:BookSources/978-0-486-47758-9"><bdi>978-0-486-47758-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fearful+Symmetry%3A+Is+God+a+Geometer%3F&rft.pages=47-48&rft.pub=Courier+Dover+Publications&rft.date=2010&rft.isbn=978-0-486-47758-9&rft.aulast=Stewart&rft.aufirst=Ian&rft.au=Golubitsky%2C+Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7x_MF83tTKQC%26pg%3DPA47&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVilfred2004" class="citation cs2">Vilfred, V. (2004), "On circulant graphs", in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J. (eds.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wG-08Lv8E-0C&pg=PA34"><i>Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001)</i></a>, Alpha Science, pp. 34–36, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/8173195692" title="Special:BookSources/8173195692"><bdi>8173195692</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+circulant+graphs&rft.btitle=Graph+Theory+and+its+Applications+%28Anna+University%2C+Chennai%2C+March+14%E2%80%9316%2C+2001%29&rft.pages=34-36&rft.pub=Alpha+Science&rft.date=2004&rft.isbn=8173195692&rft.aulast=Vilfred&rft.aufirst=V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwG-08Lv8E-0C%26pg%3DPA34&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinogradov2003" class="citation cs2"><a href="/wiki/Ivan_Matveyevich_Vinogradov" class="mw-redirect" title="Ivan Matveyevich Vinogradov">Vinogradov, I. M.</a> (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xlIfdGPM9t4C&pg=PA105">"§ VI PRIMITIVE ROOTS AND INDICES"</a>, <i>Elements of Number Theory</i>, Mineola, NY: Dover Publications, pp. 105–132, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-49530-2" title="Special:BookSources/0-486-49530-2"><bdi>0-486-49530-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A7+VI+PRIMITIVE+ROOTS+AND+INDICES&rft.btitle=Elements+of+Number+Theory&rft.place=Mineola%2C+NY&rft.pages=105-132&rft.pub=Dover+Publications&rft.date=2003&rft.isbn=0-486-49530-2&rft.aulast=Vinogradov&rft.aufirst=I.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxlIfdGPM9t4C%26pg%3DPA105&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></li></ul> <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=Cyclic_group&action=edit&section=26" 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="CITEREFHerstein1996" class="citation cs2">Herstein, I. N. (1996), <i>Abstract algebra</i> (3rd ed.), <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, pp. 53–60, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-374562-7" title="Special:BookSources/978-0-13-374562-7"><bdi>978-0-13-374562-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1375019">1375019</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+algebra&rft.pages=53-60&rft.edition=3rd&rft.pub=Prentice+Hall&rft.date=1996&rft.isbn=978-0-13-374562-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1375019%23id-name%3DMR&rft.aulast=Herstein&rft.aufirst=I.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" 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=Cyclic_group&action=edit&section=27" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Milne, Group theory, <a rel="nofollow" class="external free" href="http://www.jmilne.org/math/CourseNotes/gt.html">http://www.jmilne.org/math/CourseNotes/gt.html</a></li> <li><a rel="nofollow" class="external text" href="http://members.tripod.com/~dogschool/cyclic.html">An introduction to cyclic groups</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Cyclic_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/CyclicGroup.html">"Cyclic 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=Cyclic+Group&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCyclicGroup.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+group" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://groupnames.org/#?cyclic">Cyclic groups of small order on GroupNames</a></li> <li><a rel="nofollow" class="external text" href="https://onlinedegreemath.com/every-cyclic-group-is-abelian/">Every cyclic group is abelian</a></li></ul> <p><br /> </p> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Groups" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" 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 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<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 class="mw-selflink selflink">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 class="mw-selflink selflink">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 href="/wiki/Dihedral_group" title="Dihedral group">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> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Cyclic_group&oldid=1255668692">https://en.wikipedia.org/w/index.php?title=Cyclic_group&oldid=1255668692</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Abelian_group_theory" title="Category:Abelian group theory">Abelian group theory</a></li><li><a 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