{"id":149,"date":"2020-07-23T08:15:26","date_gmt":"2020-07-23T08:15:26","guid":{"rendered":"http:\/\/www.risanp.com\/notes\/?p=149"},"modified":"2020-07-23T08:54:54","modified_gmt":"2020-07-23T08:54:54","slug":"pinters-book-of-abstract-algebra-chapter-11-exercise-d5","status":"publish","type":"post","link":"https:\/\/www.risanp.com\/notes\/?p=149","title":{"rendered":"Pinter’s Book of Abstract Algebra Chapter 11, Exercise D5"},"content":{"rendered":"\n


Problem<\/strong>
Let \"G\" be a group and let \"a, b \in G\". Prove the following:
Let \"\operatorname{ord}(a)=n\", and suppose \"a\" has a \"k\"th root, say \"a=b^k\". Then \"\langle a\rangle=\langle b\rangle\" iff \"k\" and \"n\" are relatively prime.<\/p>\n\n\n\n

Proof.<\/strong><\/em> Consider the cyclic subgroup of \"\mathbb{Z}_{10}\". For \"a = 2\" and \"b = 1\", we have \"a = b^2\", \"n = 5\", and \"\gcd(2, 5) = 1\". However, \"\langle 1 \rangle = \mathbb{Z}_{10}\" but \"\langle 2 \rangle\" consists of all even numbers of \"\mathbb{Z}_{10}\". Hence \"\langle a \rangle \neq \langle b \rangle\". A contradiction. \"\qed\"

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ProblemLet be a group and let . Prove the following:Let , and suppose has a th root, say . Then iff and are relatively prime. Proof. Consider the cyclic subgroup of . For and , we have , , and … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,6,8],"tags":[],"class_list":["post-149","post","type-post","status-publish","format-standard","hentry","category-abstract-algebra","category-mathematics","category-pinters-book-of-abstract-algebra"],"_links":{"self":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts\/149"}],"collection":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=149"}],"version-history":[{"count":10,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts\/149\/revisions"}],"predecessor-version":[{"id":189,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts\/149\/revisions\/189"}],"wp:attachment":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}