Almost normal subgroups: Is there any notion which is weaker than normal subgroup?












7












$begingroup$


Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.



Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.






group-theory definition normal-subgroups







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
    $endgroup$
    – Eevee Trainer
    yesterday
















7












$begingroup$


Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.



Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.






group-theory definition normal-subgroups







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
    $endgroup$
    – Eevee Trainer
    yesterday














7












7








7


3



$begingroup$


Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.



Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.






group-theory definition normal-subgroups







share|cite|improve this question











$endgroup$




Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.



Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.






group-theory definition normal-subgroups




group-theory definition normal-subgroups






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 15 hours ago









Shaun

9,268113684




9,268113684










asked yesterday









I_wil_break_wallI_wil_break_wall

735




735








  • 3




    $begingroup$
    Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
    $endgroup$
    – Eevee Trainer
    yesterday














  • 3




    $begingroup$
    Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
    $endgroup$
    – Eevee Trainer
    yesterday








3




3




$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
yesterday




$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
yesterday










4 Answers
4






active

oldest

votes


















7












$begingroup$

A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.



Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.






share|cite|improve this answer











$endgroup$





















    5












    $begingroup$

    Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.






    share|cite|improve this answer









    $endgroup$





















      3












      $begingroup$

      What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.






      share|cite|improve this answer









      $endgroup$





















        2












        $begingroup$

        In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
        "For almost all $gin G$ we have $g^{-1} Ng=N$"



        In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.



        In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.






        share|cite|improve this answer











        $endgroup$













          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',
          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%2f3121058%2falmost-normal-subgroups-is-there-any-notion-which-is-weaker-than-normal-subgrou%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          7












          $begingroup$

          A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.



          Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.






          share|cite|improve this answer











          $endgroup$


















            7












            $begingroup$

            A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.



            Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.






            share|cite|improve this answer











            $endgroup$
















              7












              7








              7





              $begingroup$

              A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.



              Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.






              share|cite|improve this answer











              $endgroup$



              A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.



              Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited 15 hours ago

























              answered 21 hours ago









              Nicky HeksterNicky Hekster

              28.7k63456




              28.7k63456























                  5












                  $begingroup$

                  Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.






                  share|cite|improve this answer









                  $endgroup$


















                    5












                    $begingroup$

                    Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.






                    share|cite|improve this answer









                    $endgroup$
















                      5












                      5








                      5





                      $begingroup$

                      Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.






                      share|cite|improve this answer









                      $endgroup$



                      Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered yesterday









                      YiFanYiFan

                      4,1111627




                      4,1111627























                          3












                          $begingroup$

                          What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.






                          share|cite|improve this answer









                          $endgroup$


















                            3












                            $begingroup$

                            What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.






                            share|cite|improve this answer









                            $endgroup$
















                              3












                              3








                              3





                              $begingroup$

                              What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.






                              share|cite|improve this answer









                              $endgroup$



                              What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered yesterday









                              ThomasThomas

                              4,072510




                              4,072510























                                  2












                                  $begingroup$

                                  In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
                                  "For almost all $gin G$ we have $g^{-1} Ng=N$"



                                  In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.



                                  In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.






                                  share|cite|improve this answer











                                  $endgroup$


















                                    2












                                    $begingroup$

                                    In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
                                    "For almost all $gin G$ we have $g^{-1} Ng=N$"



                                    In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.



                                    In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.






                                    share|cite|improve this answer











                                    $endgroup$
















                                      2












                                      2








                                      2





                                      $begingroup$

                                      In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
                                      "For almost all $gin G$ we have $g^{-1} Ng=N$"



                                      In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.



                                      In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.






                                      share|cite|improve this answer











                                      $endgroup$



                                      In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
                                      "For almost all $gin G$ we have $g^{-1} Ng=N$"



                                      In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.



                                      In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.







                                      share|cite|improve this answer














                                      share|cite|improve this answer



                                      share|cite|improve this answer








                                      edited 21 hours ago

























                                      answered 23 hours ago









                                      Ali TaghaviAli Taghavi

                                      257330




                                      257330






























                                          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%2f3121058%2falmost-normal-subgroups-is-there-any-notion-which-is-weaker-than-normal-subgrou%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

                                          How to label and detect the document text images

                                          Image
                                          1 $begingroup$ This is what I mean as document text image: I want to label the texts in image as separate blocks and my model should detect these labels as classes. NOTE: This is how the end result should be like: The labels like Block 1, Block 2, Block 3,.. should be Logo, Title, Date,.. Others, etc. Work done: First approach : I tried to implement this method via Object Detection, it didn't work. It didn't even detect any text. Second approach : Then I tried it using PixelLink. As this model is build for scene text detection, it detected each and every text in the image. But this method can detect multiple lines of text if the threshold values are increased. But I have no idea how do I add labels to the text blocks. PIXEL_CLS_WEIGHT_all_ones = 'PIXEL_CLS_WEIGHT_all_ones' PIXEL_C...
                                          Read more

                                          Vallis Paradisi

                                          Image
                                          Coordinata: .mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct{display:none}.mw-parser-output .longitude,.mw-parser-output .latitude{white-space:nowrap} 33°03′S 71°37′W  /  33.05°S 71.617°W  / -33.05; -71.617 Vallis Paradisi Res apud Vicidata repertae: Civitas : Chilia Numerus incolarum : 296 655 Zona horaria : UTC-4 Situs interretialis Nomen officiale : Valparaíso Gubernium Consilium : municipal council of Valparaíso Geographia Superficies : 401 600 000 Metrum quadratum Coniunctiones urbium Urbes gemellae : Ovetum , Badalona , Santa Fe , Ora longa , Sciamhaevum , Quantunia , Callao , Bat Yam , Melaka , Novorossiysk , Busan , Corduba , Odessa , Rosario , Malacca , Barcino , Soteropolis , Franciscopolis (California) , Guanajuato, Silao Tabula aut despectus Despectus in portum sinumque et urbem Vallis Paradisi Vallis Para...
                                          Read more

                                          Tabula Rosettana

                                          Image
                                          Tabula multilinguis Rosettana in Museo Britannico ostenditur. Tabula Rosettana, [1] etiam titulo OGIS 90 agnita, est stela decreto de rebus sacris in Aegypto anno 196 a.C.n. lato inscripta. Tabula iuxta Rosettam Aegypti, urbem in delta Nili et ad oram maris Mediterranei iacentem, anno 1799 a milite Francico reperta est. Inventio stelae, linguis duabus scripturisque tribus inscriptae, eruditis Instituti Aegypti statim nuntiata est; ibi enim iussu imperatoris Napoleonis eruditi omnium scientiarum (sub aegide Commissionis Scientiarum et Artium) properaverant cum expeditione Francica. Qua a Britannis mox debellata, tabula Rosettana Londinium missa hodie apud Museum Britannicum iacet. Textus Graecus cito lectus interpretationi textuum Aegyptiorum (in formis hieroglyphica et demotica expressorum) gradatim adiuvit. Denique textum plene interpretatus est Ioannes Franciscus Champollion. Ab opere eruditorum cumulativo coepit hodiernus scripturae hieroglyphicae linguaeque Aegyptiae a...
                                          Read more