Does every subgroup of an abelian group have to be abelian?












3












$begingroup$


My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.



So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.










share|cite|improve this question











$endgroup$

















    3












    $begingroup$


    My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.



    So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.










    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$


      My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.



      So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.










      share|cite|improve this question











      $endgroup$




      My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.



      So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.







      abstract-algebra group-theory galois-theory abelian-groups






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago









      J. W. Tanner

      5,1651520




      5,1651520










      asked 6 hours ago









      MT mathMT math

      253




      253






















          2 Answers
          2






          active

          oldest

          votes


















          3












          $begingroup$

          Huh, funny, we just went over this today in my algebra class.



          Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.



          Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
            $endgroup$
            – MT math
            5 hours ago










          • $begingroup$
            I believe so, yes.
            $endgroup$
            – Eevee Trainer
            4 hours ago



















          2












          $begingroup$

          If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.






          share|cite|improve this answer









          $endgroup$














            Your Answer








            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',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            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%2f3198960%2fdoes-every-subgroup-of-an-abelian-group-have-to-be-abelian%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3












            $begingroup$

            Huh, funny, we just went over this today in my algebra class.



            Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.



            Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
              $endgroup$
              – MT math
              5 hours ago










            • $begingroup$
              I believe so, yes.
              $endgroup$
              – Eevee Trainer
              4 hours ago
















            3












            $begingroup$

            Huh, funny, we just went over this today in my algebra class.



            Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.



            Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
              $endgroup$
              – MT math
              5 hours ago










            • $begingroup$
              I believe so, yes.
              $endgroup$
              – Eevee Trainer
              4 hours ago














            3












            3








            3





            $begingroup$

            Huh, funny, we just went over this today in my algebra class.



            Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.



            Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).






            share|cite|improve this answer









            $endgroup$



            Huh, funny, we just went over this today in my algebra class.



            Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.



            Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 6 hours ago









            Eevee TrainerEevee Trainer

            10.8k31843




            10.8k31843












            • $begingroup$
              can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
              $endgroup$
              – MT math
              5 hours ago










            • $begingroup$
              I believe so, yes.
              $endgroup$
              – Eevee Trainer
              4 hours ago


















            • $begingroup$
              can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
              $endgroup$
              – MT math
              5 hours ago










            • $begingroup$
              I believe so, yes.
              $endgroup$
              – Eevee Trainer
              4 hours ago
















            $begingroup$
            can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
            $endgroup$
            – MT math
            5 hours ago




            $begingroup$
            can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
            $endgroup$
            – MT math
            5 hours ago












            $begingroup$
            I believe so, yes.
            $endgroup$
            – Eevee Trainer
            4 hours ago




            $begingroup$
            I believe so, yes.
            $endgroup$
            – Eevee Trainer
            4 hours ago











            2












            $begingroup$

            If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.






                share|cite|improve this answer









                $endgroup$



                If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 6 hours ago









                Matt SamuelMatt Samuel

                39.5k63870




                39.5k63870






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3198960%2fdoes-every-subgroup-of-an-abelian-group-have-to-be-abelian%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Knooppunt Holsloot

                    Altaar (religie)

                    Gregoriusmis