Is every normal subgroup the kernel of some self-homomorphism? [duplicate]

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This question already has an answer here:



  • Is every normal subgroup the kernel of some endomorphism?

    3 answers



Let $G$ be a group. If there is a homomorphism $f:Gto G$ (special case of the codomain being arbitrary group), then the kernel $f^-1(id)$ is a normal subgroup of $G$.



But now the other way around: Start out with the existence of a normal subgroup $H$ of $G$. Is there necessarily a homomorphism $f:Gto G$ such that the kernel of $f$ is $H$?










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marked as duplicate by Arnaud D., John Coleman, Adrian Keister, Community Mar 1 at 17:00


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    I edited the title to clarify the question. Let me know if you agree with it.
    $endgroup$
    – Dietrich Burde
    Mar 1 at 12:09










  • $begingroup$
    @DietrichBurde that’s fine thanks!
    $endgroup$
    – user56834
    Mar 1 at 12:25










  • $begingroup$
    See also math.stackexchange.com/questions/1826655/…
    $endgroup$
    – Arnaud D.
    Mar 1 at 14:09






  • 1




    $begingroup$
    A more interesting question would be: for which groups is this true? Trivially true for simple groups and also true for finite cyclic groups. Are there any others?
    $endgroup$
    – John Coleman
    Mar 1 at 14:55
















5












$begingroup$



This question already has an answer here:



  • Is every normal subgroup the kernel of some endomorphism?

    3 answers



Let $G$ be a group. If there is a homomorphism $f:Gto G$ (special case of the codomain being arbitrary group), then the kernel $f^-1(id)$ is a normal subgroup of $G$.



But now the other way around: Start out with the existence of a normal subgroup $H$ of $G$. Is there necessarily a homomorphism $f:Gto G$ such that the kernel of $f$ is $H$?










share|cite|improve this question











$endgroup$



marked as duplicate by Arnaud D., John Coleman, Adrian Keister, Community Mar 1 at 17:00


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    I edited the title to clarify the question. Let me know if you agree with it.
    $endgroup$
    – Dietrich Burde
    Mar 1 at 12:09










  • $begingroup$
    @DietrichBurde that’s fine thanks!
    $endgroup$
    – user56834
    Mar 1 at 12:25










  • $begingroup$
    See also math.stackexchange.com/questions/1826655/…
    $endgroup$
    – Arnaud D.
    Mar 1 at 14:09






  • 1




    $begingroup$
    A more interesting question would be: for which groups is this true? Trivially true for simple groups and also true for finite cyclic groups. Are there any others?
    $endgroup$
    – John Coleman
    Mar 1 at 14:55














5












5








5


1



$begingroup$



This question already has an answer here:



  • Is every normal subgroup the kernel of some endomorphism?

    3 answers



Let $G$ be a group. If there is a homomorphism $f:Gto G$ (special case of the codomain being arbitrary group), then the kernel $f^-1(id)$ is a normal subgroup of $G$.



But now the other way around: Start out with the existence of a normal subgroup $H$ of $G$. Is there necessarily a homomorphism $f:Gto G$ such that the kernel of $f$ is $H$?










share|cite|improve this question











$endgroup$





This question already has an answer here:



  • Is every normal subgroup the kernel of some endomorphism?

    3 answers



Let $G$ be a group. If there is a homomorphism $f:Gto G$ (special case of the codomain being arbitrary group), then the kernel $f^-1(id)$ is a normal subgroup of $G$.



But now the other way around: Start out with the existence of a normal subgroup $H$ of $G$. Is there necessarily a homomorphism $f:Gto G$ such that the kernel of $f$ is $H$?





This question already has an answer here:



  • Is every normal subgroup the kernel of some endomorphism?

    3 answers







group-theory normal-subgroups group-homomorphism






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 1 at 12:08









Dietrich Burde

81.5k648106




81.5k648106










asked Mar 1 at 11:55









user56834user56834

3,35921253




3,35921253




marked as duplicate by Arnaud D., John Coleman, Adrian Keister, Community Mar 1 at 17:00


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Arnaud D., John Coleman, Adrian Keister, Community Mar 1 at 17:00


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • $begingroup$
    I edited the title to clarify the question. Let me know if you agree with it.
    $endgroup$
    – Dietrich Burde
    Mar 1 at 12:09










  • $begingroup$
    @DietrichBurde that’s fine thanks!
    $endgroup$
    – user56834
    Mar 1 at 12:25










  • $begingroup$
    See also math.stackexchange.com/questions/1826655/…
    $endgroup$
    – Arnaud D.
    Mar 1 at 14:09






  • 1




    $begingroup$
    A more interesting question would be: for which groups is this true? Trivially true for simple groups and also true for finite cyclic groups. Are there any others?
    $endgroup$
    – John Coleman
    Mar 1 at 14:55

















  • $begingroup$
    I edited the title to clarify the question. Let me know if you agree with it.
    $endgroup$
    – Dietrich Burde
    Mar 1 at 12:09










  • $begingroup$
    @DietrichBurde that’s fine thanks!
    $endgroup$
    – user56834
    Mar 1 at 12:25










  • $begingroup$
    See also math.stackexchange.com/questions/1826655/…
    $endgroup$
    – Arnaud D.
    Mar 1 at 14:09






  • 1




    $begingroup$
    A more interesting question would be: for which groups is this true? Trivially true for simple groups and also true for finite cyclic groups. Are there any others?
    $endgroup$
    – John Coleman
    Mar 1 at 14:55
















$begingroup$
I edited the title to clarify the question. Let me know if you agree with it.
$endgroup$
– Dietrich Burde
Mar 1 at 12:09




$begingroup$
I edited the title to clarify the question. Let me know if you agree with it.
$endgroup$
– Dietrich Burde
Mar 1 at 12:09












$begingroup$
@DietrichBurde that’s fine thanks!
$endgroup$
– user56834
Mar 1 at 12:25




$begingroup$
@DietrichBurde that’s fine thanks!
$endgroup$
– user56834
Mar 1 at 12:25












$begingroup$
See also math.stackexchange.com/questions/1826655/…
$endgroup$
– Arnaud D.
Mar 1 at 14:09




$begingroup$
See also math.stackexchange.com/questions/1826655/…
$endgroup$
– Arnaud D.
Mar 1 at 14:09




1




1




$begingroup$
A more interesting question would be: for which groups is this true? Trivially true for simple groups and also true for finite cyclic groups. Are there any others?
$endgroup$
– John Coleman
Mar 1 at 14:55





$begingroup$
A more interesting question would be: for which groups is this true? Trivially true for simple groups and also true for finite cyclic groups. Are there any others?
$endgroup$
– John Coleman
Mar 1 at 14:55











1 Answer
1






active

oldest

votes


















7












$begingroup$

That's false.



If $G=mathbbZ$ then any homomorphism $f:mathbbZrightarrowmathbbZ$ takes the form $f(a)=ma$ for some $minmathbbZ$. Clearly the kernel is trivial (unless $m=0$ then the kernel is everything). However for all $ninmathbbN$ we have that $nmathbbZ$ is a normal subgroup of $mathbbZ$. In particular $2mathbbZ$ is not a kernel of any homomorphim from $mathbbZ$ to itself.



However, it is possible to "correct" this statement. If you only request an homomorphism from $G$ to some other group. Given any normal subgroup $H$, the quotient homomorphism $f:Grightarrow G/H$ which sends $gmapsto g+H$ has $H$ as it's kernel. In other words, every normal subgroup is a kernel of some homomorphism, not necessarily from $G$ to itself.






share|cite|improve this answer









$endgroup$



















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    7












    $begingroup$

    That's false.



    If $G=mathbbZ$ then any homomorphism $f:mathbbZrightarrowmathbbZ$ takes the form $f(a)=ma$ for some $minmathbbZ$. Clearly the kernel is trivial (unless $m=0$ then the kernel is everything). However for all $ninmathbbN$ we have that $nmathbbZ$ is a normal subgroup of $mathbbZ$. In particular $2mathbbZ$ is not a kernel of any homomorphim from $mathbbZ$ to itself.



    However, it is possible to "correct" this statement. If you only request an homomorphism from $G$ to some other group. Given any normal subgroup $H$, the quotient homomorphism $f:Grightarrow G/H$ which sends $gmapsto g+H$ has $H$ as it's kernel. In other words, every normal subgroup is a kernel of some homomorphism, not necessarily from $G$ to itself.






    share|cite|improve this answer









    $endgroup$

















      7












      $begingroup$

      That's false.



      If $G=mathbbZ$ then any homomorphism $f:mathbbZrightarrowmathbbZ$ takes the form $f(a)=ma$ for some $minmathbbZ$. Clearly the kernel is trivial (unless $m=0$ then the kernel is everything). However for all $ninmathbbN$ we have that $nmathbbZ$ is a normal subgroup of $mathbbZ$. In particular $2mathbbZ$ is not a kernel of any homomorphim from $mathbbZ$ to itself.



      However, it is possible to "correct" this statement. If you only request an homomorphism from $G$ to some other group. Given any normal subgroup $H$, the quotient homomorphism $f:Grightarrow G/H$ which sends $gmapsto g+H$ has $H$ as it's kernel. In other words, every normal subgroup is a kernel of some homomorphism, not necessarily from $G$ to itself.






      share|cite|improve this answer









      $endgroup$















        7












        7








        7





        $begingroup$

        That's false.



        If $G=mathbbZ$ then any homomorphism $f:mathbbZrightarrowmathbbZ$ takes the form $f(a)=ma$ for some $minmathbbZ$. Clearly the kernel is trivial (unless $m=0$ then the kernel is everything). However for all $ninmathbbN$ we have that $nmathbbZ$ is a normal subgroup of $mathbbZ$. In particular $2mathbbZ$ is not a kernel of any homomorphim from $mathbbZ$ to itself.



        However, it is possible to "correct" this statement. If you only request an homomorphism from $G$ to some other group. Given any normal subgroup $H$, the quotient homomorphism $f:Grightarrow G/H$ which sends $gmapsto g+H$ has $H$ as it's kernel. In other words, every normal subgroup is a kernel of some homomorphism, not necessarily from $G$ to itself.






        share|cite|improve this answer









        $endgroup$



        That's false.



        If $G=mathbbZ$ then any homomorphism $f:mathbbZrightarrowmathbbZ$ takes the form $f(a)=ma$ for some $minmathbbZ$. Clearly the kernel is trivial (unless $m=0$ then the kernel is everything). However for all $ninmathbbN$ we have that $nmathbbZ$ is a normal subgroup of $mathbbZ$. In particular $2mathbbZ$ is not a kernel of any homomorphim from $mathbbZ$ to itself.



        However, it is possible to "correct" this statement. If you only request an homomorphism from $G$ to some other group. Given any normal subgroup $H$, the quotient homomorphism $f:Grightarrow G/H$ which sends $gmapsto g+H$ has $H$ as it's kernel. In other words, every normal subgroup is a kernel of some homomorphism, not necessarily from $G$ to itself.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 1 at 11:59









        YankoYanko

        8,1282830




        8,1282830












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