{"id":186095,"date":"2025-01-24T06:16:31","date_gmt":"2025-01-24T06:16:31","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186095"},"modified":"2025-01-24T06:16:34","modified_gmt":"2025-01-24T06:16:34","slug":"write-the-cayley-table-for-the-dihedral-group-d4-with-8-elements","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/24\/write-the-cayley-table-for-the-dihedral-group-d4-with-8-elements\/","title":{"rendered":"Write the Cayley table for the Dihedral group D4 with 8 elements"},"content":{"rendered":"\n<p>Write the Cayley table for the Dihedral group D4 with 8 elements.<\/p>\n\n\n\n<p>Why is D8 not isomorphic to Z\/8Z?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cayley Table for the Dihedral Group D4D_4:<\/h3>\n\n\n\n<p>The Dihedral group D4D_4 represents the symmetries of a square and has 8 elements: four rotations and four reflections. The elements of D4D_4 can be represented as:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>r0r^0 (the identity rotation),<\/li>\n\n\n\n<li>r1r^1, r2r^2, r3r^3 (rotations by 90\u00b0, 180\u00b0, and 270\u00b0 respectively),<\/li>\n\n\n\n<li>s1s^1, s2s^2, s3s^3, s4s^4 (reflections across axes).<\/li>\n<\/ul>\n\n\n\n<p>The Cayley table for D4D_4 is as follows, with the elements r0,r1,r2,r3r^0, r^1, r^2, r^3 for rotations and s1,s2,s3,s4s^1, s^2, s^3, s^4 for reflections:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>*<\/th><th>r0r^0<\/th><th>r1r^1<\/th><th>r2r^2<\/th><th>r3r^3<\/th><th>s1s^1<\/th><th>s2s^2<\/th><th>s3s^3<\/th><th>s4s^4<\/th><\/tr><\/thead><tbody><tr><td>r0r^0<\/td><td>r0r^0<\/td><td>r1r^1<\/td><td>r2r^2<\/td><td>r3r^3<\/td><td>s1s^1<\/td><td>s2s^2<\/td><td>s3s^3<\/td><td>s4s^4<\/td><\/tr><tr><td>r1r^1<\/td><td>r1r^1<\/td><td>r2r^2<\/td><td>r3r^3<\/td><td>r0r^0<\/td><td>s2s^2<\/td><td>s3s^3<\/td><td>s4s^4<\/td><td>s1s^1<\/td><\/tr><tr><td>r2r^2<\/td><td>r2r^2<\/td><td>r3r^3<\/td><td>r0r^0<\/td><td>r1r^1<\/td><td>s3s^3<\/td><td>s4s^4<\/td><td>s1s^1<\/td><td>s2s^2<\/td><\/tr><tr><td>r3r^3<\/td><td>r3r^3<\/td><td>r0r^0<\/td><td>r1r^1<\/td><td>r2r^2<\/td><td>s4s^4<\/td><td>s1s^1<\/td><td>s2s^2<\/td><td>s3s^3<\/td><\/tr><tr><td>s1s^1<\/td><td>s1s^1<\/td><td>s4s^4<\/td><td>s3s^3<\/td><td>s2s^2<\/td><td>r1r^1<\/td><td>r2r^2<\/td><td>r3r^3<\/td><td>r0r^0<\/td><\/tr><tr><td>s2s^2<\/td><td>s2s^2<\/td><td>s3s^3<\/td><td>s4s^4<\/td><td>s1s^1<\/td><td>r2r^2<\/td><td>r3r^3<\/td><td>r0r^0<\/td><td>r1r^1<\/td><\/tr><tr><td>s3s^3<\/td><td>s3s^3<\/td><td>s2s^2<\/td><td>s1s^1<\/td><td>s4s^4<\/td><td>r3r^3<\/td><td>r0r^0<\/td><td>r1r^1<\/td><td>r2r^2<\/td><\/tr><tr><td>s4s^4<\/td><td>s4s^4<\/td><td>s1s^1<\/td><td>s2s^2<\/td><td>s3s^3<\/td><td>r0r^0<\/td><td>r1r^1<\/td><td>r2r^2<\/td><td>r3r^3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Why is D8D_8 Not Isomorphic to Z\/8Z\\mathbb{Z}\/8\\mathbb{Z}?<\/h3>\n\n\n\n<p>D8D_8, the Dihedral group of order 8, and Z\/8Z\\mathbb{Z}\/8\\mathbb{Z}, the cyclic group of order 8, are not isomorphic due to their structural differences. The two key reasons for this are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Cyclic vs. Non-Cyclic Structure:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Z\/8Z\\mathbb{Z}\/8\\mathbb{Z} is a cyclic group. This means that there exists an element, specifically the generator 1, such that every other element in the group can be written as a power of this generator. In other words, Z\/8Z\\mathbb{Z}\/8\\mathbb{Z} has an element of order 8, which generates the entire group.<\/li>\n\n\n\n<li>D8D_8, on the other hand, is not cyclic. It is generated by two elements: a rotation (say, rr) and a reflection (say, ss). The rotation elements in D8D_8 only generate a subgroup of order 4, while the reflections generate a subgroup of order 2. There is no element in D8D_8 that has order 8. This non-cyclic nature of D8D_8 directly implies that it cannot be isomorphic to the cyclic group Z\/8Z\\mathbb{Z}\/8\\mathbb{Z}.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Element Orders:<\/strong>\n<ul class=\"wp-block-list\">\n<li>In Z\/8Z\\mathbb{Z}\/8\\mathbb{Z}, the element 1 has order 8, meaning that repeated addition of 1 (mod 8) will eventually cover all elements of the group.<\/li>\n\n\n\n<li>In D8D_8, the highest order of any element is 4 (from the rotations). The elements in D8D_8 have orders 1, 2, and 4, and there is no element with order 8, unlike in Z\/8Z\\mathbb{Z}\/8\\mathbb{Z}.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p>Therefore, the structural differences in element orders and the cyclic versus non-cyclic nature of these groups prevent D8D_8 from being isomorphic to Z\/8Z\\mathbb{Z}\/8\\mathbb{Z}.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Write the Cayley table for the Dihedral group D4 with 8 elements. Why is D8 not isomorphic to Z\/8Z? The Correct Answer and Explanation is : Cayley Table for the Dihedral Group D4D_4: The Dihedral group D4D_4 represents the symmetries of a square and has 8 elements: four rotations and four reflections. The elements of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-186095","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186095","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=186095"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186095\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186095"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}