{"id":186097,"date":"2025-01-24T06:19:40","date_gmt":"2025-01-24T06:19:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186097"},"modified":"2025-01-24T06:19:42","modified_gmt":"2025-01-24T06:19:42","slug":"the-cayley-table-for-the-dihedral-group-d4-of-order-8-is-given-below","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/24\/the-cayley-table-for-the-dihedral-group-d4-of-order-8-is-given-below\/","title":{"rendered":"The Cayley table for the dihedral group (D4,) of order 8 is given below"},"content":{"rendered":"\n

The Cayley table for the dihedral group (D4,) of order 8 is given below, where we write zo y as just zy for all z, y \u2208 Da R R90 R180 R270 H V D D Ro Ro R90 R180 R270 H V D D’ R90 R00 R180 R270 Ro D’ D H V R180 R180 R170 R170 R210 Ro R V H D D R Roo R D D’ V H H H D V D R R Roo R210 Ro V V D’ H DD D V D’ D’ H DHV DD R R270 R270 R90 Ro R Ro R R R 1a. State the identity of the group: en = D. 1b. Determine if H is a solution of the equation zen, or zz = ep 1c. Find all elements of the set {zz D\u2081 and z\u00b2 = en).<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

1a. Identity of the group<\/h3>\n\n\n\n

The dihedral group ( D_4 ) is the group of symmetries of a square, which consists of both rotations and reflections. The order of this group is 8, as the table indicates there are 8 elements. These elements can be thought of as:<\/p>\n\n\n\n

    \n
  • ( R_0 ) (identity rotation by 0 degrees)<\/li>\n\n\n\n
  • ( R_{90} ) (rotation by 90 degrees)<\/li>\n\n\n\n
  • ( R_{180} ) (rotation by 180 degrees)<\/li>\n\n\n\n
  • ( R_{270} ) (rotation by 270 degrees)<\/li>\n\n\n\n
  • ( H ) (horizontal reflection)<\/li>\n\n\n\n
  • ( V ) (vertical reflection)<\/li>\n\n\n\n
  • ( D ) (diagonal reflection 1)<\/li>\n\n\n\n
  • ( D’ ) (diagonal reflection 2)<\/li>\n<\/ul>\n\n\n\n

    In a group, the identity element is the one that leaves all other elements unchanged when combined with them. The identity element of ( D_4 ) is ( R_0 ), because:<\/p>\n\n\n\n

      \n
    • ( R_0 ) combined with any element ( x \\in D_4 ) (either by composition or multiplication) gives ( x ) back, i.e., ( R_0 \\cdot x = x \\cdot R_0 = x ).<\/li>\n<\/ul>\n\n\n\n

      Thus, the identity element is ( R_0 ).<\/p>\n\n\n\n

      1b. Checking if ( H ) is a solution to ( z \\cdot e = e \\cdot p )<\/h3>\n\n\n\n

      Let’s check if ( H ) (the horizontal reflection) satisfies the equation ( z \\cdot e = e \\cdot p ), where ( e ) is the identity element (which is ( R_0 ) here).<\/p>\n\n\n\n

        \n
      • The equation ( z \\cdot e = e \\cdot p ) essentially asks if multiplying the element ( H ) by the identity element ( R_0 ) results in the same element, i.e., does ( H \\cdot R_0 = R_0 \\cdot H )?<\/li>\n\n\n\n
      • Since ( R_0 ) is the identity, the equation holds trivially because for any element ( x ), ( R_0 \\cdot x = x \\cdot R_0 = x ).<\/li>\n<\/ul>\n\n\n\n

        Thus, ( H ) is indeed a solution to ( z \\cdot e = e \\cdot p ).<\/p>\n\n\n\n

        1c. Finding elements where ( z^2 = R_0 ) (identity)<\/h3>\n\n\n\n

        To find the elements ( z ) in ( D_4 ) such that ( z^2 = R_0 ), we need to check which elements in the group satisfy this property. We will square each element and check the result:<\/p>\n\n\n\n

          \n
        • ( R_0^2 = R_0 ) (identity)<\/li>\n\n\n\n
        • ( R_{90}^2 = R_{180} )<\/li>\n\n\n\n
        • ( R_{180}^2 = R_0 ) (identity)<\/li>\n\n\n\n
        • ( R_{270}^2 = R_{180} )<\/li>\n\n\n\n
        • ( H^2 = R_0 ) (identity)<\/li>\n\n\n\n
        • ( V^2 = R_0 ) (identity)<\/li>\n\n\n\n
        • ( D^2 = R_0 ) (identity)<\/li>\n\n\n\n
        • ( D’^2 = R_0 ) (identity)<\/li>\n<\/ul>\n\n\n\n

          Thus, the elements ( z \\in D_4 ) that satisfy ( z^2 = R_0 ) are:<\/p>\n\n\n\n

            \n
          • ( R_0 ) (identity)<\/li>\n\n\n\n
          • ( R_{180} )<\/li>\n\n\n\n
          • ( H )<\/li>\n\n\n\n
          • ( V )<\/li>\n\n\n\n
          • ( D )<\/li>\n\n\n\n
          • ( D’ )<\/li>\n<\/ul>\n\n\n\n

            Conclusion<\/h3>\n\n\n\n
              \n
            1. The identity element of ( D_4 ) is ( R_0 ), the identity rotation.<\/strong><\/li>\n\n\n\n
            2. ( H ) does satisfy the equation ( z \\cdot e = e \\cdot p ) because ( R_0 ) is the identity element.<\/strong><\/li>\n\n\n\n
            3. The elements where ( z^2 = R_0 ) are ( R_0 ), ( R_{180} ), ( H ), ( V ), ( D ), and ( D’ ).<\/strong><\/li>\n<\/ol>\n\n\n\n

              These observations are crucial in understanding the structure of the dihedral group ( D_4 ), which encapsulates both rotational and reflectional symmetries of a square. The relations between elements in the group are foundational for studying symmetries in geometry and algebra.<\/p>\n","protected":false},"excerpt":{"rendered":"

              The Cayley table for the dihedral group (D4,) of order 8 is given below, where we write zo y as just zy for all z, y \u2208 Da R R90 R180 R270 H V D D Ro Ro R90 R180 R270 H V D D’ R90 R00 R180 R270 Ro D’ D H V R180 […]<\/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-186097","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186097","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=186097"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186097\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186097"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186097"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186097"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}