The hookrightarrow and its meaning












3












$begingroup$


Very weirdly I found a $hookrightarrow $ in my Algebra book just like that. Could someone please just tell me what it means? Sorry for this maybe stupid question in advance but very much appreciate your help!






abstract-algebra notation







share|cite|improve this question











$endgroup$








  • 11




    $begingroup$
    Most probably, it denotes an injective morphism.
    $endgroup$
    – Bernard
    5 hours ago










  • $begingroup$
    It's used between groups at this point... pretty weird it wasn't explained before using it.
    $endgroup$
    – KingDingeling
    5 hours ago










  • $begingroup$
    Between groups, I'm sure.
    $endgroup$
    – Bernard
    5 hours ago






  • 4




    $begingroup$
    $Ahookrightarrow B$ primarily means the usual inclusion if $Asubseteq B$, i.e. the restriction of the identity function. More generally, it can also denote arbitrary embedding, which, in algebraic context, is usually the same as 'injective homomorphism'.
    $endgroup$
    – Berci
    4 hours ago












  • $begingroup$
    thanks guys for your help! much appreciated here :)
    $endgroup$
    – KingDingeling
    4 hours ago
















3












$begingroup$


Very weirdly I found a $hookrightarrow $ in my Algebra book just like that. Could someone please just tell me what it means? Sorry for this maybe stupid question in advance but very much appreciate your help!






abstract-algebra notation







share|cite|improve this question











$endgroup$








  • 11




    $begingroup$
    Most probably, it denotes an injective morphism.
    $endgroup$
    – Bernard
    5 hours ago










  • $begingroup$
    It's used between groups at this point... pretty weird it wasn't explained before using it.
    $endgroup$
    – KingDingeling
    5 hours ago










  • $begingroup$
    Between groups, I'm sure.
    $endgroup$
    – Bernard
    5 hours ago






  • 4




    $begingroup$
    $Ahookrightarrow B$ primarily means the usual inclusion if $Asubseteq B$, i.e. the restriction of the identity function. More generally, it can also denote arbitrary embedding, which, in algebraic context, is usually the same as 'injective homomorphism'.
    $endgroup$
    – Berci
    4 hours ago












  • $begingroup$
    thanks guys for your help! much appreciated here :)
    $endgroup$
    – KingDingeling
    4 hours ago














3












3








3





$begingroup$


Very weirdly I found a $hookrightarrow $ in my Algebra book just like that. Could someone please just tell me what it means? Sorry for this maybe stupid question in advance but very much appreciate your help!






abstract-algebra notation







share|cite|improve this question











$endgroup$




Very weirdly I found a $hookrightarrow $ in my Algebra book just like that. Could someone please just tell me what it means? Sorry for this maybe stupid question in advance but very much appreciate your help!






abstract-algebra notation




abstract-algebra notation






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

3,2701320




3,2701320










asked 5 hours ago









KingDingelingKingDingeling

1837




1837








  • 11




    $begingroup$
    Most probably, it denotes an injective morphism.
    $endgroup$
    – Bernard
    5 hours ago










  • $begingroup$
    It's used between groups at this point... pretty weird it wasn't explained before using it.
    $endgroup$
    – KingDingeling
    5 hours ago










  • $begingroup$
    Between groups, I'm sure.
    $endgroup$
    – Bernard
    5 hours ago






  • 4




    $begingroup$
    $Ahookrightarrow B$ primarily means the usual inclusion if $Asubseteq B$, i.e. the restriction of the identity function. More generally, it can also denote arbitrary embedding, which, in algebraic context, is usually the same as 'injective homomorphism'.
    $endgroup$
    – Berci
    4 hours ago












  • $begingroup$
    thanks guys for your help! much appreciated here :)
    $endgroup$
    – KingDingeling
    4 hours ago














  • 11




    $begingroup$
    Most probably, it denotes an injective morphism.
    $endgroup$
    – Bernard
    5 hours ago










  • $begingroup$
    It's used between groups at this point... pretty weird it wasn't explained before using it.
    $endgroup$
    – KingDingeling
    5 hours ago










  • $begingroup$
    Between groups, I'm sure.
    $endgroup$
    – Bernard
    5 hours ago






  • 4




    $begingroup$
    $Ahookrightarrow B$ primarily means the usual inclusion if $Asubseteq B$, i.e. the restriction of the identity function. More generally, it can also denote arbitrary embedding, which, in algebraic context, is usually the same as 'injective homomorphism'.
    $endgroup$
    – Berci
    4 hours ago












  • $begingroup$
    thanks guys for your help! much appreciated here :)
    $endgroup$
    – KingDingeling
    4 hours ago








11




11




$begingroup$
Most probably, it denotes an injective morphism.
$endgroup$
– Bernard
5 hours ago




$begingroup$
Most probably, it denotes an injective morphism.
$endgroup$
– Bernard
5 hours ago












$begingroup$
It's used between groups at this point... pretty weird it wasn't explained before using it.
$endgroup$
– KingDingeling
5 hours ago




$begingroup$
It's used between groups at this point... pretty weird it wasn't explained before using it.
$endgroup$
– KingDingeling
5 hours ago












$begingroup$
Between groups, I'm sure.
$endgroup$
– Bernard
5 hours ago




$begingroup$
Between groups, I'm sure.
$endgroup$
– Bernard
5 hours ago




4




4




$begingroup$
$Ahookrightarrow B$ primarily means the usual inclusion if $Asubseteq B$, i.e. the restriction of the identity function. More generally, it can also denote arbitrary embedding, which, in algebraic context, is usually the same as 'injective homomorphism'.
$endgroup$
– Berci
4 hours ago






$begingroup$
$Ahookrightarrow B$ primarily means the usual inclusion if $Asubseteq B$, i.e. the restriction of the identity function. More generally, it can also denote arbitrary embedding, which, in algebraic context, is usually the same as 'injective homomorphism'.
$endgroup$
– Berci
4 hours ago














$begingroup$
thanks guys for your help! much appreciated here :)
$endgroup$
– KingDingeling
4 hours ago




$begingroup$
thanks guys for your help! much appreciated here :)
$endgroup$
– KingDingeling
4 hours ago










2 Answers
2






active

oldest

votes


















3












$begingroup$

Resolved in comments, so I'll make this community wiki. $F hookrightarrow G$ denotes the inclusion of $F$ in $G$, with $hookrightarrow$ denoting a monomorphism.






share|cite|improve this answer











$endgroup$





















    1












    $begingroup$

    In an algebraic setting, $hookrightarrow$ denotes a monic map and $twoheadrightarrow$ denotes an epic map.






    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%2f3149834%2fthe-hookrightarrow-and-its-meaning%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












      $begingroup$

      Resolved in comments, so I'll make this community wiki. $F hookrightarrow G$ denotes the inclusion of $F$ in $G$, with $hookrightarrow$ denoting a monomorphism.






      share|cite|improve this answer











      $endgroup$


















        3












        $begingroup$

        Resolved in comments, so I'll make this community wiki. $F hookrightarrow G$ denotes the inclusion of $F$ in $G$, with $hookrightarrow$ denoting a monomorphism.






        share|cite|improve this answer











        $endgroup$
















          3












          3








          3





          $begingroup$

          Resolved in comments, so I'll make this community wiki. $F hookrightarrow G$ denotes the inclusion of $F$ in $G$, with $hookrightarrow$ denoting a monomorphism.






          share|cite|improve this answer











          $endgroup$



          Resolved in comments, so I'll make this community wiki. $F hookrightarrow G$ denotes the inclusion of $F$ in $G$, with $hookrightarrow$ denoting a monomorphism.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          answered 4 hours ago


























          community wiki





          bounceback
























              1












              $begingroup$

              In an algebraic setting, $hookrightarrow$ denotes a monic map and $twoheadrightarrow$ denotes an epic map.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                In an algebraic setting, $hookrightarrow$ denotes a monic map and $twoheadrightarrow$ denotes an epic map.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  In an algebraic setting, $hookrightarrow$ denotes a monic map and $twoheadrightarrow$ denotes an epic map.






                  share|cite|improve this answer









                  $endgroup$



                  In an algebraic setting, $hookrightarrow$ denotes a monic map and $twoheadrightarrow$ denotes an epic map.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 4 hours ago









                  MPWMPW

                  30.7k12157




                  30.7k12157






























                      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%2f3149834%2fthe-hookrightarrow-and-its-meaning%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

                      الفوسفات في المغرب

                      Image
                      قطار ينقل فوسفات من منطقة خريبكة متجه نحو الدار البيضاء. ديار عمال بمصانع الفوسفات بمدينة آسفي. المحتوى هنا ينقصه الاستشهاد بمصادر. يرجى إيراد مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (أغسطس 2015) الفوسفات في المغرب هو ثالث مصدر عالمي للفوسفات، ويتوفر المغرب على %75 من الاحتياطي العالمي  [بحاجة لمصدر] ويتوزع بنسب متفاوتة بين عدة مناطق أساسية وهي: وادي زم بن جرير بوكراع خريبكة اليوسفية أولاد عبدون كنتور واد الذهب مستقلة. هذه الأخيرة لم تفتح بعد للاستغلال. الصادرات الحامض الفوسفوري P205 تمتص الهند %50 من ا لصادرات الإجمالية آسيا (دون روسيا)%59 أوروبا %30 أمريكا اللاتينية %11 أسمدة صلبة أوروبا 38 % أمريكا اللاتينية %30 آسيا (دون روسيا) 22 % أستراليا 9 % بوابة المغرب هذه بذرة مقالة عن علم الاقتصاد أو موضوع متعلق به بحاجة للتوسيع. شارك في تحريرها. This page is only for reference, If you need detailed information, please check here
                      Read more

                      Four equal circles intersect: What is the area of the small shaded portion and its height

                      Image
                      3 $begingroup$ In terms of $R$ which is the radius of all four circles, what is the area of the intersection region of these four equal circles and the height of the marked arrow in the figure? The marked arrow is along the line CD, also the midpoint of all the circles are points A, B, C and D. Looking for a very short intuitive solution. I have checked similar questions on this site for example this and this. geometry circle share | cite | improve this question edited 2 hours ago Abdulhameed asked 2 hours ago ...
                      Read more

                      جامعة ليفربول

                      Image
                      جامعة ليفربول University of Liverpool الأسماء السابقة كلية ليفربول الجامعية (1882-1884) جامعة فيكتوريا في ليفربول (1884-1903) معلومات التأسيس 1882 (منذ 137 سنة) الإنتماءات رابطة الجامعات الأوروبية مجموعة راسل جامعات القرميد الأحمر المنحة المالية 143.5 مليون £ النوع جامعة عامة الكليات 6 الموقع الجغرافي إحداثيات 53°24′22″N 2°58′01″W  /  53.406°N 2.967°W  / 53.406; -2.967   المدينة ليفربول المكان   إنجلترا البلد   المملكة المتحدة رقم الهاتف 7942000 151 (44+) إحصاءات الطلاب 17,405 (سنة 2008) عضوية مجموعة رسل [1] أورسيد [2]   طلاب الدراسات العليا 3,860 الموقع www.liverpool.ac.uk تعديل   جامعة ليفربول مبنى فكتوريا جامعة ليفربول (بالإنجليزية: The University of Liverpool )) إحدى الجامعات البريطانية العريقة، تقع في مدينة ليفربول على الساحل الشمالي الغربي من أنجلترا . وهي عضو في مجموعة راسل للجامعات وتعتبر من ضمن جامعات الطوب الأحمر الستة. محتويات ...
                      Read more