Euler totient function and groups.

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Say $phi(n)=k$ is the number of integers less than n relatively prime to n. Then prove that any integer a relatively prime to n $a^phi(n)=1 quad modquad n$.



My proof



U(n) be the group of integers coprime to n less than n. Then a belongs to this group and order of group is k. Hence $a^k=1$.



This proof is correct right.










share|cite|improve this question

























    up vote
    0
    down vote

    favorite












    Say $phi(n)=k$ is the number of integers less than n relatively prime to n. Then prove that any integer a relatively prime to n $a^phi(n)=1 quad modquad n$.



    My proof



    U(n) be the group of integers coprime to n less than n. Then a belongs to this group and order of group is k. Hence $a^k=1$.



    This proof is correct right.










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Say $phi(n)=k$ is the number of integers less than n relatively prime to n. Then prove that any integer a relatively prime to n $a^phi(n)=1 quad modquad n$.



      My proof



      U(n) be the group of integers coprime to n less than n. Then a belongs to this group and order of group is k. Hence $a^k=1$.



      This proof is correct right.










      share|cite|improve this question













      Say $phi(n)=k$ is the number of integers less than n relatively prime to n. Then prove that any integer a relatively prime to n $a^phi(n)=1 quad modquad n$.



      My proof



      U(n) be the group of integers coprime to n less than n. Then a belongs to this group and order of group is k. Hence $a^k=1$.



      This proof is correct right.







      finite-groups totient-function






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Sep 10 at 8:45









      Piyush Divyanakar

      3,315222




      3,315222




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote













          If a (finite) group has order $k$ then of course every element $x$ of the group satisfies $x^k=1$; this is pure group theory.



          However to apply this theorem one has to first show that U(n) is a group in the first place.
          SO one has to do some number-theoretic proof, there is no way to avoid it.






          share|cite|improve this answer




















            Your Answer




            StackExchange.ifUsing("editor", function ()
            return StackExchange.using("mathjaxEditing", function ()
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            );
            );
            , "mathjax-editing");

            StackExchange.ready(function()
            var channelOptions =
            tags: "".split(" "),
            id: "69"
            ;
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function()
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled)
            StackExchange.using("snippets", function()
            createEditor();
            );

            else
            createEditor();

            );

            function createEditor()
            StackExchange.prepareEditor(
            heartbeatType: 'answer',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );













             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2911693%2feuler-totient-function-and-groups%23new-answer', 'question_page');

            );

            Post as a guest






























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            If a (finite) group has order $k$ then of course every element $x$ of the group satisfies $x^k=1$; this is pure group theory.



            However to apply this theorem one has to first show that U(n) is a group in the first place.
            SO one has to do some number-theoretic proof, there is no way to avoid it.






            share|cite|improve this answer
























              up vote
              1
              down vote













              If a (finite) group has order $k$ then of course every element $x$ of the group satisfies $x^k=1$; this is pure group theory.



              However to apply this theorem one has to first show that U(n) is a group in the first place.
              SO one has to do some number-theoretic proof, there is no way to avoid it.






              share|cite|improve this answer






















                up vote
                1
                down vote










                up vote
                1
                down vote









                If a (finite) group has order $k$ then of course every element $x$ of the group satisfies $x^k=1$; this is pure group theory.



                However to apply this theorem one has to first show that U(n) is a group in the first place.
                SO one has to do some number-theoretic proof, there is no way to avoid it.






                share|cite|improve this answer












                If a (finite) group has order $k$ then of course every element $x$ of the group satisfies $x^k=1$; this is pure group theory.



                However to apply this theorem one has to first show that U(n) is a group in the first place.
                SO one has to do some number-theoretic proof, there is no way to avoid it.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Sep 10 at 9:05









                P Vanchinathan

                14.2k12036




                14.2k12036



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2911693%2feuler-totient-function-and-groups%23new-answer', 'question_page');

                    );

                    Post as a guest













































































                    這個網誌中的熱門文章

                    Why am i infinitely getting the same tweet with the Twitter Search API?

                    Is there any way to eliminate the singular point to solve this integral by hand or by approximations?

                    Strongly p-embedded subgroups and p-Sylow subgroups.