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/* cache key: grouppropssubwiki:resourceloader:filter:minify-js:7:f11cf079be51878ff699b1c46f5476c7 */ }</script> <script>if(window.mw){ mw.loader.load(["mediawiki.page.startup","mediawiki.legacy.wikibits","mediawiki.legacy.ajax","skins.vector.js"]); }</script> <link rel="alternate" type="application/rdf+xml" title="Cyclic group:Z3" href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Special:ExportRDF/Cyclic_group:Z3&xmlmime=rdf"/> </head> <body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-Cyclic_group_Z3 skin-vector action-view vector-animateLayout"> <div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="mw-js-message" style="display:none;"></div> <h1 id="firstHeading" class="firstHeading" lang="en"><span dir="auto">Cyclic group:Z3</span></h1> <div id="bodyContent" class="mw-body-content"> <div id="siteSub">From Groupprops</div> <div id="contentSub"></div> <div id="jump-to-nav" class="mw-jump"> Jump to: <a href="#mw-navigation">navigation</a>, <a href="#p-search">search</a> </div> <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><blockquote class="toccolours" style="float:none; background:white; padding: 10px 15px 10px 15px; display:table;"> This article is about a particular <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Group" title="Group">group</a>, i.e., a group unique upto <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Isomorphism_of_groups" title="Isomorphism of groups">isomorphism</a>. <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5BSpecific-20information-20about::Cyclic-20group:Z3-5D-5D" title="Special:Ask/-5B-5BSpecific-20information-20about::Cyclic-20group:Z3-5D-5D">View specific information (such as linear representation theory, subgroup structure) about this group</a> <br/><b><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Category:Particular_groups" title="Category:Particular groups">View a complete list of particular groups</a> (this is a very huge list!)</b><a id="toggledisplay1l" href="javascript:toggleDisplay( "toggledisplay1", "[HIDE]", "[SHOW MORE]" )" style="font-size:larger">[SHOW MORE]</a><div id="toggledisplay1" style="display:none;"></div></blockquote> <div id="toc" class="toc"><div id="toctitle"><h2>Contents</h2></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Verbal_definition"><span class="tocnumber">1.1</span> <span class="toctext">Verbal definition</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Multiplication_table"><span class="tocnumber">1.2</span> <span class="toctext">Multiplication table</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-4"><a href="#Arithmetic_functions"><span class="tocnumber">2</span> <span class="toctext">Arithmetic functions</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Group_properties"><span class="tocnumber">3</span> <span class="toctext">Group properties</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Endomorphisms"><span class="tocnumber">4</span> <span class="toctext">Endomorphisms</span></a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Automorphisms"><span class="tocnumber">4.1</span> <span class="toctext">Automorphisms</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Subgroups"><span class="tocnumber">5</span> <span class="toctext">Subgroups</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Quotients"><span class="tocnumber">6</span> <span class="toctext">Quotients</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Other_constructions"><span class="tocnumber">7</span> <span class="toctext">Other constructions</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#In_larger_groups"><span class="tocnumber">8</span> <span class="toctext">In larger groups</span></a> <ul> <li class="toclevel-2 tocsection-12"><a href="#Occurrence_as_a_subgroup"><span class="tocnumber">8.1</span> <span class="toctext">Occurrence as a subgroup</span></a></li> <li class="toclevel-2 tocsection-13"><a href="#Occurrence_as_a_normal_subgroup"><span class="tocnumber">8.2</span> <span class="toctext">Occurrence as a normal subgroup</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-14"><a href="#GAP_implementation"><span class="tocnumber">9</span> <span class="toctext">GAP implementation</span></a> <ul> <li class="toclevel-2"><a href="#Group_ID"><span class="tocnumber">9.1</span> <span class="toctext">Group ID</span></a></li> <li class="toclevel-2 tocsection-15"><a href="#Other_descriptions"><span class="tocnumber">9.2</span> <span class="toctext">Other descriptions</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-16"><a href="#Internal_links"><span class="tocnumber">10</span> <span class="toctext">Internal links</span></a></li> </ul> </div> <h2><span class="mw-headline" id="Definition">Definition</span></h2> <h3><span class="mw-headline" id="Verbal_definition">Verbal definition</span></h3> <p>The cyclic group of order 3 is defined as the unique group of order 3. Equivalently it can be described as a group with three elements <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/b/b/3/bb3a6b570bbc41dcacee93de685b09e3.png"/> where <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/b/d/2/bd22a3f4edadf900e4bbc95967d63516.png"/> with the exponent reduced mod 3. It can also be viewed as: </p> <ul> <li> The <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Quotient_group" title="Quotient group" class="mw-redirect">quotient group</a> of the group of integers by the subgroup of multiples of 3 </li> <li> The multiplicative group comprising the three cuberoots of unity (as a subgroup of the multiplicative group of nonzero complex numbers) </li> <li> The <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Alternating_group" title="Alternating group">alternating group</a> on three elements </li> <li> The group of orientation-preserving symmetries (rotational symmetries) of the equilateral triangle </li> <li> The <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Multiplicative_group_of_a_field" title="Multiplicative group of a field">multiplicative group</a> of the field of four elements. In particular, it is the <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/General_linear_group" title="General linear group" class="mw-redirect">general linear group</a> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/1/3/9/1390135a5ab7a1aef69365c264cf242f.png"/>. </li> </ul> <h3><span class="mw-headline" id="Multiplication_table">Multiplication table</span></h3> <table class="wikitable" border="1"> <tr> <th> Element </th> <th> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/> (<a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Identity_element" title="Identity element" class="mw-redirect">identity element</a>) </th> <th> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/> (generator) </th> <th> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/> (generator) </th></tr> <tr> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/> </td></tr> <tr> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/> </td></tr> <tr> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/> </td> <td> <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/> </td></tr></table> <h2><span class="mw-headline" id="Arithmetic_functions">Arithmetic functions</span></h2> <blockquote class="toccolours" style="float:none; background:white; padding: 10px 15px 10px 15px; display:table;"> Want to compare and contrast arithmetic function values with other groups of the same <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Order_of_a_group" title="Order of a group">order</a>? Check out <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Groups_of_order_3#Arithmetic_functions" title="Groups of order 3">groups of order 3#Arithmetic functions</a></blockquote> <table class="sortable" border="1"> <tr> <th> Function </th> <th> Value </th> <th> Similar groups </th> <th> Explanation for function value </th></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Underlying_prime_of_p-group" title="Underlying prime of p-group">underlying prime of p-group</a> </td> <td> 3 </td> <td> </td> <td> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Order_of_a_group" title="Order of a group">order</a> (number of elements, equivalently, cardinality or size of underlying set) </td> <td> 3 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order</a> </td> <td> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Prime-base_logarithm_of_order" title="Prime-base logarithm of order">prime-base logarithm of order</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order</a> </td> <td> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Max-length_of_a_group" title="Max-length of a group">max-length of a group</a> </td> <td> 1 </td> <td> </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Max-length_of_a_group" title="Max-length of a group">max-length of a group</a> equals <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Prime-base_logarithm_of_order" title="Prime-base logarithm of order">prime-base logarithm of order</a> for <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Group_of_prime_power_order" title="Group of prime power order">group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Chief_length&action=edit&redlink=1" class="new" title="Chief length (page does not exist)">chief length</a> </td> <td> 1 </td> <td> </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Chief_length&action=edit&redlink=1" class="new" title="Chief length (page does not exist)">chief length</a> equals <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Prime-base_logarithm_of_order" title="Prime-base logarithm of order">prime-base logarithm of order</a> for <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Group_of_prime_power_order" title="Group of prime power order">group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Composition_length" title="Composition length">composition length</a> </td> <td> 1 </td> <td> </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Composition_length" title="Composition length">composition length</a> equals <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Prime-base_logarithm_of_order" title="Prime-base logarithm of order">prime-base logarithm of order</a> for <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Group_of_prime_power_order" title="Group of prime power order">group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Exponent_of_a_group" title="Exponent of a group">exponent of a group</a> </td> <td> 3 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::exponent-20of-20a-20group;3-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::exponent-20of-20a-20group;3-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and exponent of a group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::exponent-20of-20a-20group;3-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::exponent-20of-20a-20group;3-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and exponent of a group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::exponent-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::exponent-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same exponent of a group</a> </td> <td> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Prime-base_logarithm_of_exponent" title="Prime-base logarithm of exponent">prime-base logarithm of exponent</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20exponent;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20exponent;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and prime-base logarithm of exponent</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20exponent;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20exponent;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and prime-base logarithm of exponent</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20exponent;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20exponent;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of exponent</a> </td> <td> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Frattini_length" title="Frattini length">Frattini length</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::Frattini-20length;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::Frattini-20length;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and Frattini length</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::Frattini-20length;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::Frattini-20length;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and Frattini length</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::Frattini-20length;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::Frattini-20length;1-5D-5D/-3FGAP-20ID">groups with same Frattini length</a> </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Frattini_length" title="Frattini length">Frattini length</a> equals <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Prime-base_logarithm_of_exponent" title="Prime-base logarithm of exponent">prime-base logarithm of exponent</a> for <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group_of_prime_power_order" title="Abelian group of prime power order">abelian group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Minimum_size_of_generating_set" title="Minimum size of generating set">minimum size of generating set</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::minimum-20size-20of-20generating-20set;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::minimum-20size-20of-20generating-20set;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and minimum size of generating set</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::minimum-20size-20of-20generating-20set;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::minimum-20size-20of-20generating-20set;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and minimum size of generating set</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::minimum-20size-20of-20generating-20set;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::minimum-20size-20of-20generating-20set;1-5D-5D/-3FGAP-20ID">groups with same minimum size of generating set</a> </td> <td> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Subgroup_rank_of_a_group" title="Subgroup rank of a group">subgroup rank of a group</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::subgroup-20rank-20of-20a-20group;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::subgroup-20rank-20of-20a-20group;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and subgroup rank of a group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::subgroup-20rank-20of-20a-20group;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::subgroup-20rank-20of-20a-20group;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and subgroup rank of a group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::subgroup-20rank-20of-20a-20group;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::subgroup-20rank-20of-20a-20group;1-5D-5D/-3FGAP-20ID">groups with same subgroup rank of a group</a> </td> <td> same as minimum size of generating set since it is an <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group_of_prime_power_order" title="Abelian group of prime power order">abelian group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Rank_of_a_p-group" title="Rank of a p-group">rank of a p-group</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and rank of a p-group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and rank of a p-group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::rank-20of-20a-20p-2Dgroup;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::rank-20of-20a-20p-2Dgroup;1-5D-5D/-3FGAP-20ID">groups with same rank of a p-group</a> </td> <td> same as minimum size of generating set since it is an <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group_of_prime_power_order" title="Abelian group of prime power order">abelian group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Normal_rank_of_a_p-group" title="Normal rank of a p-group">normal rank of a p-group</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::normal-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::normal-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and normal rank of a p-group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::normal-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::normal-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and normal rank of a p-group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::normal-20rank-20of-20a-20p-2Dgroup;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::normal-20rank-20of-20a-20p-2Dgroup;1-5D-5D/-3FGAP-20ID">groups with same normal rank of a p-group</a> </td> <td> same as minimum size of generating set since it is an <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group_of_prime_power_order" title="Abelian group of prime power order">abelian group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Characteristic_rank_of_a_p-group" title="Characteristic rank of a p-group">characteristic rank of a p-group</a> </td> <td> 1 </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::characteristic-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::characteristic-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::order-20of-20a-20group;3-5D-5D/-3FGAP-20ID">groups with same order and characteristic rank of a p-group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::characteristic-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::characteristic-20rank-20of-20a-20p-2Dgroup;1-5D-5D-5B-5Barithmetic-20function-20value::prime-2Dbase-20logarithm-20of-20order;1-5D-5D/-3FGAP-20ID">groups with same prime-base logarithm of order and characteristic rank of a p-group</a> | <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Ask/-5B-5Barithmetic-20function-20value::characteristic-20rank-20of-20a-20p-2Dgroup;1-5D-5D/-3FGAP-20ID" title="Special:Ask/-5B-5Barithmetic-20function-20value::characteristic-20rank-20of-20a-20p-2Dgroup;1-5D-5D/-3FGAP-20ID">groups with same characteristic rank of a p-group</a> </td> <td> same as minimum size of generating set since it is an <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group_of_prime_power_order" title="Abelian group of prime power order">abelian group of prime power order</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Nilpotency_class" title="Nilpotency class">nilpotency class</a> </td> <td> 1 </td> <td> </td> <td> The group is a nontrivial <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group" title="Abelian group">abelian group</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Derived_length" title="Derived length">derived length</a> </td> <td> 1 </td> <td> </td> <td> The group is a nontrivial <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group" title="Abelian group">abelian group</a> </td></tr> <tr> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Fitting_length" title="Fitting length">Fitting length</a> </td> <td> 1 </td> <td> </td> <td> The group is a nontrivial <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group" title="Abelian group">abelian group</a> <p>- </p> </td></tr></table> <h2><span class="mw-headline" id="Group_properties">Group properties</span></h2> <table class="wikitable" border="1"> <tr> <th>Property </th> <th> Satisfied </th> <th> Explanation </th> <th> Comment </th></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Group_of_prime_order" title="Group of prime order">Group of prime order</a> </td> <td> Yes </td> <td> By definition </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> </td> <td> Yes </td> <td> By definition </td> <td> Smallest <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Odd-order_cyclic_group" title="Odd-order cyclic group">odd-order cyclic group</a> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a> </td> <td> Yes </td> <td> By definition </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_group" title="Abelian group">Abelian group</a> </td> <td> Yes </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Cyclic_implies_abelian" title="Cyclic implies abelian">Cyclic implies abelian</a> </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Nilpotent_group" title="Nilpotent group">Nilpotent group</a> </td> <td> Yes </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Abelian_implies_nilpotent" title="Abelian implies nilpotent">Abelian implies nilpotent</a> </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Metacyclic_group" title="Metacyclic group">Metacyclic group</a> </td> <td> Yes </td> <td> Cyclic implies metacyclic </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Supersolvable_group" title="Supersolvable group">Supersolvable group</a> </td> <td> Yes </td> <td> Cyclic implies supersolvable </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Solvable_group" title="Solvable group">Solvable group</a> </td> <td> Yes </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Abelian_implies_solvable&action=edit&redlink=1" class="new" title="Abelian implies solvable (page does not exist)">Abelian implies solvable</a> </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/T-group" title="T-group">T-group</a> </td> <td> Yes </td> <td> </td> <td> </td></tr> <tr> <td><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Simple_group" title="Simple group">Simple group</a> </td> <td> Yes </td> <td> <a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Equivalence_of_definitions_of_group_of_prime_order&action=edit&redlink=1" class="new" title="Equivalence of definitions of group of prime order (page does not exist)">Simple abelian if and only if prime order</a> </td> <td> </td></tr></table> <h2><span class="mw-headline" id="Endomorphisms">Endomorphisms</span></h2> <p>Any endomorphism of a cyclic group is determined by where it sends the generator. The cyclic group of order three has three endomorphisms: </p> <ul> <li> The identity map is an endomorphism. This map sends every element to itself. </li> <li> The square map is an endomorphism. This map sends <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/> to <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/>, <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/c/6/6/c66452631491acdbf8e5ed69dfd19681.png"/> to <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png"/>, and <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/> to itself. </li> <li> The trivial map is an endomorphism. This map sends every element to <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/e/1/6/e1671797c52e15f763380b45e841ec32.png"/>. </li> </ul> <h3><span class="mw-headline" id="Automorphisms">Automorphisms</span></h3> <p>Of the three endomorphisms, two are automorphisms: the identity map and the square map. These form a cyclic group of order two: the square map, applied twice, gives the identity map. </p> <h2><span class="mw-headline" id="Subgroups">Subgroups</span></h2> <p>The group has no proper nontrivial subgroups: the only subgroups are the whole group and the trivial subgroup. </p><p>More generally, any nontrivial group with no proper nontrivial subgroup must be cyclic of prime order. Conversely, any cyclic group of prime order has no proper nontrivial subgroup. <tt>Further information: <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/No_proper_nontrivial_subgroup_implies_cyclic_of_prime_order" title="No proper nontrivial subgroup implies cyclic of prime order">No proper nontrivial subgroup implies cyclic of prime order</a></tt> </p> <h2><span class="mw-headline" id="Quotients">Quotients</span></h2> <p>The cyclic group of order three has only two quotients: the whole group and the trivial quotient. This follows from the fact that this group is simple -- it has no proper nontrivial normal subgroup. </p> <h2><span class="mw-headline" id="Other_constructions">Other constructions</span></h2> <p><a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Template:Holomorph&action=edit&redlink=1" class="new" title="Template:Holomorph (page does not exist)">Template:Holomorph</a> </p><p>The holomorph of this group, i.e., the semidirect product of this group with its automorphism group, is isomorphic to the symmetric group on three letters. The cyclic group sits inside this as the alternating group and the automorphism group sits inside as a subgroup of order two. </p> <h2><span class="mw-headline" id="In_larger_groups">In larger groups</span></h2> <h3><span class="mw-headline" id="Occurrence_as_a_subgroup">Occurrence as a subgroup</span></h3> <p>The cyclic group of order 3 occurs as a subgroup in many groups. In general, a group contains a cyclic subgroup of order three if and only if its order is a multiple of three (this follows from <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Cauchy%27s_theorem" title="Cauchy's theorem">Cauchy's theorem</a>, a corollary of <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Sylow%27s_theorem" title="Sylow's theorem">Sylow's theorem</a>). </p> <h3><span class="mw-headline" id="Occurrence_as_a_normal_subgroup">Occurrence as a normal subgroup</span></h3> <p>The cyclic group of order three occurs as a normal subgroup in some groups. </p><p>For instance, if a field contains non-identity cuberoots of unity, then the multiplicative group of the field contains a cyclic subgroup of order three. As a corollary, the general linear group contains a central subgroup of order three. </p><p>Normal subgroups of order three need not be central; for instance, in the <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Symmetric_group:S3" title="Symmetric group:S3">symmetric group on three letters</a>, the alternating group is a normal subgroup of order three, but is not central. However, for an <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Odd-order_group" title="Odd-order group">odd-order group</a>, any normal subgroup of order three is central. This follows from a more general fact: <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Normal_of_least_prime_order_implies_central" title="Normal of least prime order implies central" class="mw-redirect">a normal subgroup whose order is the least prime divisor of the order of the group is central</a>. </p> <h2><span class="mw-headline" id="GAP_implementation">GAP implementation</span></h2> <h3><span class="mw-headline" id="Group_ID">Group ID</span></h3> <p>This <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Finite_group" title="Finite group">finite group</a> has <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Groups_of_order_3" title="Groups of order 3">order 3</a> and has ID 1 among the groups of order 3 in GAP's SmallGroup library. For context, there are groups of order 3. It can thus be defined using GAP's <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/GAP:SmallGroup" title="GAP:SmallGroup">SmallGroup</a> function as: </p><p><tt>SmallGroup(3,1)</tt> </p><p>For instance, we can use the following assignment in GAP to create the group and name it <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/d/f/c/dfcf28d0734569a6a693bc8194de62bf.png"/>: </p><p><tt>gap> G := SmallGroup(3,1);</tt> </p><p>Conversely, to check whether a given group <img class="tex" alt="" src="/web/20150701221056im_/http://groupprops.subwiki.org/w/images/math/d/f/c/dfcf28d0734569a6a693bc8194de62bf.png"/> is in fact the group we want, we can use GAP's <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/GAP:IdGroup" title="GAP:IdGroup">IdGroup</a> function: </p><p><tt>IdGroup(G) = [3,1]</tt> </p><p>or just do: </p><p><tt>IdGroup(G)</tt> </p><p>to have GAP output the group ID, that we can then compare to what we want. </p><p><br/> </p> <h3><span class="mw-headline" id="Other_descriptions">Other descriptions</span></h3> <p>The cyclic group of order three can be defined using the <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/GAP:CyclicGroup" title="GAP:CyclicGroup">CyclicGroup</a> function: </p><p><tt>CyclicGroup(3)</tt> </p><p>It can also be defined as the <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Alternating_group" title="Alternating group">alternating group</a> of degree three, using the <a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=GAP:AlternatingGroup&action=edit&redlink=1" class="new" title="GAP:AlternatingGroup (page does not exist)">AlternatingGroup</a> function: </p><p><tt>AlternatingGroup(3)</tt> </p> <h2><span class="mw-headline" id="Internal_links">Internal links</span></h2> <ul> <li> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_cyclic_group:Z3" title="Linear representation theory of cyclic group:Z3">Linear representation theory of cyclic group:Z3</a> </li> <li> <a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Galois_extensions_for_cyclic_group:Z3" title="Galois extensions for cyclic group:Z3">Galois extensions for cyclic group:Z3</a> </li> <li> <a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Group_cohomology_of_cyclic_group:Z3&action=edit&redlink=1" class="new" title="Group cohomology of cyclic group:Z3 (page does not exist)">Group cohomology of cyclic group:Z3</a> </li> </ul>   </div> <div class="printfooter"> Retrieved from "<a dir="ltr" href="https://web.archive.org/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Cyclic_group:Z3&oldid=44276">http://groupprops.subwiki.org/w/index.php?title=Cyclic_group:Z3&oldid=44276</a>" </div> <div id="catlinks" class="catlinks"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Special:Categories" title="Special:Categories">Category</a>: <ul><li><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Category:Particular_groups" title="Category:Particular groups">Particular groups</a></li></ul></div></div> <div class="visualClear"></div> </div> </div> <div id="mw-navigation"> <h2>Navigation menu</h2> <div id="mw-head"> <div id="p-personal" role="navigation" class="" aria-labelledby="p-personal-label"> <h3 id="p-personal-label">Personal tools</h3> <ul> <li id="pt-login"><a href="/web/20150701221056/http://groupprops.subwiki.org/w/index.php?title=Special:UserLogin&returnto=Cyclic+group%3AZ3" title="You are encouraged to log in; 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href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Dihedral_group:D8">Dihedral group:D8 (order 8)</a></li> <li id="n-Symmetric-group:S5-.28order-5.21-.3D-120.29"><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Symmetric_group:S5">Symmetric group:S5 (order 5! = 120)</a></li> <li id="n-Alternating-group:A5-.28order-5.21.2F2-.3D-60.29"><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Alternating_group:A5">Alternating group:A5 (order 5!/2 = 60)</a></li> <li id="n-Quaternion-group-.28order-8.29"><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Quaternion_group">Quaternion group (order 8)</a></li> <li id="n-All-particular-groups-.28long-list.29"><a href="/web/20150701221056/http://groupprops.subwiki.org/wiki/Category:Particular_groups">All particular groups (long list)</a></li> </ul> </div> </div> <div class="portal" role="navigation" id="p-tb" aria-labelledby="p-tb-label"> <h3 id="p-tb-label">Tools</h3> <div class="body"> <ul> <li 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