{"id":830,"date":"2013-01-23T09:57:37","date_gmt":"2013-01-23T09:57:37","guid":{"rendered":"http:\/\/www.faultorfeature.com\/?p=830"},"modified":"2013-01-23T09:57:37","modified_gmt":"2013-01-23T09:57:37","slug":"prove-that-the-products-ab-and-ba-are-conjugate-elements-in-a-group","status":"publish","type":"post","link":"https:\/\/www.faultorfeature.com\/?p=830","title":{"rendered":"prove that the products ab and ba are conjugate elements in a group?"},"content":{"rendered":"<p><strong><i>Question by Jenny H<\/i>: prove that the products ab and ba are conjugate elements in a group?<\/strong><\/p>\n<p>prove that the products ab and ba are conjugate elements in a group. Please show steps!<\/p>\n<p><strong>Best answer:<\/strong><\/p>\n<p><i>Answer by Copestone<\/i><br \/>The answer is almost tirivial.  It is just a two line proof but since you ask for details, I would write down everything for you, including definitions.<\/p>\n<p>Let a, b ? G, a group.  Let e be the identity element.<\/p>\n<p>Recall from definition that the products ab and ba are conjugates if there is an element x ? G such that<br \/>\nx(ab) = (ba)x<\/p>\n<p>So, simply take x = a^{-1}<\/p>\n<p>Then x(ab) = (a^{-1})(ab) = (a^{-1})(a) = eb = b<br \/>\nwhile (ba)x = (ba)(a^{-1}) = b((a)(a^{-1}) = be = b.<\/p>\n<p>It follows that x(ab) = (ba)x.<br \/>\nEquivalently, some textbook also writes x(ab)x^{-1} = ba.<\/p>\n<p><strong>Give your answer to this question below!<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Question by Jenny H: prove that the products ab and ba are conjugate elements in a group? prove that the products ab and ba are conjugate elements in a group. Please show steps! Best answer: Answer by CopestoneThe answer is almost tirivial. It is just a two line proof but since you ask for details, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[2192,2193,2194,204,2191],"class_list":["post-830","post","type-post","status-publish","format-standard","hentry","category-recalls","tag-conjugate","tag-elements","tag-group","tag-products","tag-prove"],"_links":{"self":[{"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=\/wp\/v2\/posts\/830","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=830"}],"version-history":[{"count":0,"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=\/wp\/v2\/posts\/830\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=830"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=830"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.faultorfeature.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=830"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}