Let $phi:Bbb Z_nrightarrow G$ s.t. $phi(i)=h^i$ for $0le ile n$. Give necessary and sufficient condition for...












2












$begingroup$


The exercise reads




Let $G$ be a group, $h$ and element of $G$, and $n$ a positive integer. Let $phi : mathbb{Z}_nrightarrow G$ be defined by $phi(i)=h^i$ for $0leq ileq n$. Give a necessary and sufficient condition (in terms of $h$ and $n$) for $phi$ to be a homomorphism. Prove your assertion.




I always have problems with necessary and sufficient arguments, because I do not know how to prove. I know it implies an if and only if, but I'm not sure exactly over which is used. Now, if I look at the answer, it says:




The map is a homomorphism if and only if $h^n=e$, the identity in $G$.




But how do you know that I have to prove $h^n=e$?



I would have thought that I have to prove $phi(n+m)=phi(n)phi(m)$, so it's quite confusing to think that we have to prove $h^n=e$.






group-theory group-homomorphism







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Could it be that the question is asking about $phi:Bbb Z_n to G$?
    $endgroup$
    – Omnomnomnom
    Dec 21 '17 at 18:27






  • 1




    $begingroup$
    Missing an $n$ in $mathbb{Z}$.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:30






  • 3




    $begingroup$
    Obvious for you.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:40






  • 2




    $begingroup$
    So maybe start by proving that $tilde{phi}: mathbb{Z}rightarrow G$, $i mapsto h^i$ is always a homomorphism.
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:27








  • 2




    $begingroup$
    After that, think about a condition on $tilde{phi}$ that would allow you to replace $mathbb{Z}$ with $mathbb{Z}/nmathbb{Z}$ (hint: first isomorphism theorem).
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:31
















2












$begingroup$


The exercise reads




Let $G$ be a group, $h$ and element of $G$, and $n$ a positive integer. Let $phi : mathbb{Z}_nrightarrow G$ be defined by $phi(i)=h^i$ for $0leq ileq n$. Give a necessary and sufficient condition (in terms of $h$ and $n$) for $phi$ to be a homomorphism. Prove your assertion.




I always have problems with necessary and sufficient arguments, because I do not know how to prove. I know it implies an if and only if, but I'm not sure exactly over which is used. Now, if I look at the answer, it says:




The map is a homomorphism if and only if $h^n=e$, the identity in $G$.




But how do you know that I have to prove $h^n=e$?



I would have thought that I have to prove $phi(n+m)=phi(n)phi(m)$, so it's quite confusing to think that we have to prove $h^n=e$.






group-theory group-homomorphism







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Could it be that the question is asking about $phi:Bbb Z_n to G$?
    $endgroup$
    – Omnomnomnom
    Dec 21 '17 at 18:27






  • 1




    $begingroup$
    Missing an $n$ in $mathbb{Z}$.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:30






  • 3




    $begingroup$
    Obvious for you.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:40






  • 2




    $begingroup$
    So maybe start by proving that $tilde{phi}: mathbb{Z}rightarrow G$, $i mapsto h^i$ is always a homomorphism.
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:27








  • 2




    $begingroup$
    After that, think about a condition on $tilde{phi}$ that would allow you to replace $mathbb{Z}$ with $mathbb{Z}/nmathbb{Z}$ (hint: first isomorphism theorem).
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:31














2












2








2





$begingroup$


The exercise reads




Let $G$ be a group, $h$ and element of $G$, and $n$ a positive integer. Let $phi : mathbb{Z}_nrightarrow G$ be defined by $phi(i)=h^i$ for $0leq ileq n$. Give a necessary and sufficient condition (in terms of $h$ and $n$) for $phi$ to be a homomorphism. Prove your assertion.




I always have problems with necessary and sufficient arguments, because I do not know how to prove. I know it implies an if and only if, but I'm not sure exactly over which is used. Now, if I look at the answer, it says:




The map is a homomorphism if and only if $h^n=e$, the identity in $G$.




But how do you know that I have to prove $h^n=e$?



I would have thought that I have to prove $phi(n+m)=phi(n)phi(m)$, so it's quite confusing to think that we have to prove $h^n=e$.






group-theory group-homomorphism







share|cite|improve this question











$endgroup$




The exercise reads




Let $G$ be a group, $h$ and element of $G$, and $n$ a positive integer. Let $phi : mathbb{Z}_nrightarrow G$ be defined by $phi(i)=h^i$ for $0leq ileq n$. Give a necessary and sufficient condition (in terms of $h$ and $n$) for $phi$ to be a homomorphism. Prove your assertion.




I always have problems with necessary and sufficient arguments, because I do not know how to prove. I know it implies an if and only if, but I'm not sure exactly over which is used. Now, if I look at the answer, it says:




The map is a homomorphism if and only if $h^n=e$, the identity in $G$.




But how do you know that I have to prove $h^n=e$?



I would have thought that I have to prove $phi(n+m)=phi(n)phi(m)$, so it's quite confusing to think that we have to prove $h^n=e$.






group-theory group-homomorphism




group-theory group-homomorphism






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 22 '18 at 11:37









Shaun

9,814113684




9,814113684










asked Dec 21 '17 at 18:22









user2820579user2820579

782417




782417








  • 1




    $begingroup$
    Could it be that the question is asking about $phi:Bbb Z_n to G$?
    $endgroup$
    – Omnomnomnom
    Dec 21 '17 at 18:27






  • 1




    $begingroup$
    Missing an $n$ in $mathbb{Z}$.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:30






  • 3




    $begingroup$
    Obvious for you.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:40






  • 2




    $begingroup$
    So maybe start by proving that $tilde{phi}: mathbb{Z}rightarrow G$, $i mapsto h^i$ is always a homomorphism.
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:27








  • 2




    $begingroup$
    After that, think about a condition on $tilde{phi}$ that would allow you to replace $mathbb{Z}$ with $mathbb{Z}/nmathbb{Z}$ (hint: first isomorphism theorem).
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:31














  • 1




    $begingroup$
    Could it be that the question is asking about $phi:Bbb Z_n to G$?
    $endgroup$
    – Omnomnomnom
    Dec 21 '17 at 18:27






  • 1




    $begingroup$
    Missing an $n$ in $mathbb{Z}$.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:30






  • 3




    $begingroup$
    Obvious for you.
    $endgroup$
    – user2820579
    Dec 21 '17 at 18:40






  • 2




    $begingroup$
    So maybe start by proving that $tilde{phi}: mathbb{Z}rightarrow G$, $i mapsto h^i$ is always a homomorphism.
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:27








  • 2




    $begingroup$
    After that, think about a condition on $tilde{phi}$ that would allow you to replace $mathbb{Z}$ with $mathbb{Z}/nmathbb{Z}$ (hint: first isomorphism theorem).
    $endgroup$
    – Tim kinsella
    Dec 21 '17 at 19:31








1




1




$begingroup$
Could it be that the question is asking about $phi:Bbb Z_n to G$?
$endgroup$
– Omnomnomnom
Dec 21 '17 at 18:27




$begingroup$
Could it be that the question is asking about $phi:Bbb Z_n to G$?
$endgroup$
– Omnomnomnom
Dec 21 '17 at 18:27




1




1




$begingroup$
Missing an $n$ in $mathbb{Z}$.
$endgroup$
– user2820579
Dec 21 '17 at 18:30




$begingroup$
Missing an $n$ in $mathbb{Z}$.
$endgroup$
– user2820579
Dec 21 '17 at 18:30




3




3




$begingroup$
Obvious for you.
$endgroup$
– user2820579
Dec 21 '17 at 18:40




$begingroup$
Obvious for you.
$endgroup$
– user2820579
Dec 21 '17 at 18:40




2




2




$begingroup$
So maybe start by proving that $tilde{phi}: mathbb{Z}rightarrow G$, $i mapsto h^i$ is always a homomorphism.
$endgroup$
– Tim kinsella
Dec 21 '17 at 19:27






$begingroup$
So maybe start by proving that $tilde{phi}: mathbb{Z}rightarrow G$, $i mapsto h^i$ is always a homomorphism.
$endgroup$
– Tim kinsella
Dec 21 '17 at 19:27






2




2




$begingroup$
After that, think about a condition on $tilde{phi}$ that would allow you to replace $mathbb{Z}$ with $mathbb{Z}/nmathbb{Z}$ (hint: first isomorphism theorem).
$endgroup$
– Tim kinsella
Dec 21 '17 at 19:31




$begingroup$
After that, think about a condition on $tilde{phi}$ that would allow you to replace $mathbb{Z}$ with $mathbb{Z}/nmathbb{Z}$ (hint: first isomorphism theorem).
$endgroup$
– Tim kinsella
Dec 21 '17 at 19:31










1 Answer
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$begingroup$

This community wiki answer is to point out that this comment followed by this comment, both posted above by @Timkinsella (who is invited to post his own answer), form an answer to the question.



Summarising them:



1) Prove $tilde{phi}: Bbb Zto G, imapsto h^i$ is always a homomorphism.



2) Think of the condition on $tilde{phi}$ that would allow $Bbb Z$ to be replaced by $Bbb Z/nBbb Z$.




Hint: First isomorphism theorem.







share|cite|improve this answer











$endgroup$














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    1 Answer
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    active

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    active

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    active

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    2












    $begingroup$

    This community wiki answer is to point out that this comment followed by this comment, both posted above by @Timkinsella (who is invited to post his own answer), form an answer to the question.



    Summarising them:



    1) Prove $tilde{phi}: Bbb Zto G, imapsto h^i$ is always a homomorphism.



    2) Think of the condition on $tilde{phi}$ that would allow $Bbb Z$ to be replaced by $Bbb Z/nBbb Z$.




    Hint: First isomorphism theorem.







    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      This community wiki answer is to point out that this comment followed by this comment, both posted above by @Timkinsella (who is invited to post his own answer), form an answer to the question.



      Summarising them:



      1) Prove $tilde{phi}: Bbb Zto G, imapsto h^i$ is always a homomorphism.



      2) Think of the condition on $tilde{phi}$ that would allow $Bbb Z$ to be replaced by $Bbb Z/nBbb Z$.




      Hint: First isomorphism theorem.







      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        This community wiki answer is to point out that this comment followed by this comment, both posted above by @Timkinsella (who is invited to post his own answer), form an answer to the question.



        Summarising them:



        1) Prove $tilde{phi}: Bbb Zto G, imapsto h^i$ is always a homomorphism.



        2) Think of the condition on $tilde{phi}$ that would allow $Bbb Z$ to be replaced by $Bbb Z/nBbb Z$.




        Hint: First isomorphism theorem.







        share|cite|improve this answer











        $endgroup$



        This community wiki answer is to point out that this comment followed by this comment, both posted above by @Timkinsella (who is invited to post his own answer), form an answer to the question.



        Summarising them:



        1) Prove $tilde{phi}: Bbb Zto G, imapsto h^i$ is always a homomorphism.



        2) Think of the condition on $tilde{phi}$ that would allow $Bbb Z$ to be replaced by $Bbb Z/nBbb Z$.




        Hint: First isomorphism theorem.








        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        answered Dec 22 '18 at 11:29


























        community wiki





        Shaun































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