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

                      Liste der Baudenkmale in Friedland (Mecklenburg)

                      Image
                      In der Liste der Baudenkmale in Friedland sind alle denkmalgeschützten Bauten der Stadt Friedland (Mecklenburg-Vorpommern) und ihrer Ortsteile aufgelistet. Grundlage ist die Veröffentlichung der Denkmalliste des Landkreises Mecklenburgische Seenplatte (Auszug) vom 28. September 2016. Inhaltsverzeichnis 1 Baudenkmale nach Ortsteilen 1.1 Friedland 1.2 Bresewitz 1.3 Brohm 1.4 Cosa 1.5 Dishley 1.6 Dishley-Ausbau 1.7 Eichhorst 1.8 Glienke 1.9 Heinrichshöh 1.10 Heinrichswalde 1.11 Jatzke 1.12 Jatzke-Meierei 1.13 Liepen 1.14 Ramelow 1.15 Schwanbeck 2 Vorgesehener Denkmalbereich 3 Quelle 4 Weblinks Baudenkmale nach Ortsteilen | Friedland | ID Lage Bezeichnung Beschreibung Bild 281 An der Kleinbahn 2 Windmühle mit Umgang 282 Anklamer Straße alter Friedhof mit Portalwand 282 Anklamer Straße drei Erbbegräbnissen und Gräbern u. a. H.A. Riemann 284
                      Read more

                      Single-Malt-Whisky

                      Image
                      Fünf verschiedene schottische Single-Malt-Whiskys Als Single-Malt-Whisky oder kürzer Single Malt werden Whiskys bezeichnet, die zwei besondere Bedingungen erfüllen: Sie stammen zum ersten aus einer einzigen Brennerei, sind also kein Verschnitt aus mehreren Whiskysorten (daher Single , deutsch: einzeln). Zum zweiten wird als Getreide ausschließlich gemälzte Gerste verwendet (deswegen Malt , deutsch: Malz). Der Zusatz Single Malt wird als Prädikat verstanden, die so ausgezeichneten Whiskys sind in aller Regel besonders hochwertig. Hinzu kommt ihre besonders lange Lagerung, die Reifungszeit währt oft ein Jahrzehnt und länger. Inhaltsverzeichnis 1 Allgemeines 2 Herkunftsländer 2.1 Schottland 2.2 Irland 2.3 Japan 2.4 Kontinentaleuropa 3 Produktionsschritte 3.1 Mälzen 3.2 Brauen 3.3 Brennen 3.4 Reifung und Finish 3.5 Abfüllung 4 Altersangaben 5 Lagerung von Whiskyflaschen 6 Geschmacksklassifikation 7 Sonstig
                      Read more

                      Czorneboh

                      Image
                      Dieser Artikel erläutert den Berg Czorneboh, für die Gottheit siehe Czorneboh (Gottheit). Czorneboh Čornobóh Berggasthof mit Turm Höhe 555,7  m ü. NHN [1] Lage Freistaat Sachsen, Deutschland Gebirge Lausitzer Bergland Dominanz 15,4 km →  Kottmar Koordinaten 51° 7′ 11″  N , 14° 31′ 31″  O 51.119722222222 14.525194444444 555.7 Koordinaten: 51° 7′ 11″  N , 14° 31′ 31″  O Gestein Granodiorit Besonderheiten Aussichtsturm Czorneboh Der Czorneboh , obersorbisch Čornobóh , ist ein Berg zwischen Hochkirch und Cunewalde in der Oberlausitz. Er gehört zu der südöstlich von Bautzen gelegenen Czorneboh-Bergkette und ist mit einer Höhe von 555,7  m ü. NHN [1] der höchste Punkt dieses Ausläufers des Lausitzer Berglandes. Er leitet als Landschaftselement die wellige Granitlandschaft der Lausitzer Vorbergzone ein. Der Gipfel des Czorneboh liegt in der Gemarkung Meschwitz (Gemeinde Hochkirch), die Czornebohbaude jedoch
                      Read more