Prove the following permutation mapping is a homomorphism.












0












$begingroup$


Prove that $varphi: S_n to S_{n+1 }$, where for $sigmain S_{n}$: begin{pmatrix} 1 & 2 & dots & n \ 1sigma & 2sigma & dots & nsigma\ end{pmatrix}



$varphi(sigma)in S_{n+1}= begin{pmatrix} 1 & 2 & dots & n & n+1 \ 1sigma & 2sigma & dots & nsigma & (n+1)sigma \ end{pmatrix} $
is a homomorphism.



Wondering if anyone can explain how to prove this. Thanks.




From the comments:




I literally have no idea where to start, hence why I haven't given details of where ive got stuck as I have no idea what methods I should use to even start it. I have done a few questions similar using matrices, trigonometric functions and other functions and found them fine. I have been using the standard definition for homomorphism that is that φ(ab)=φ(a)φ(b). However, with this question this did not seem to get me very far at all and I was wondering if there was something more clever or a silly detail I was overlooking.







permutations group-homomorphism







share|cite|improve this question











$endgroup$












  • $begingroup$
    How have you proved this?
    $endgroup$
    – Finn Johnson
    Dec 31 '18 at 15:23






  • 1




    $begingroup$
    Welcome to MSE @FinnJohnson. Your question may attract some negative votes. I let you here why it can be like that: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc."
    $endgroup$
    – idriskameni
    Dec 31 '18 at 15:30








  • 2




    $begingroup$
    If you want answers, at the very least don't post questions as if you were instructing us to do what you want.
    $endgroup$
    – José Carlos Santos
    Dec 31 '18 at 15:30






  • 2




    $begingroup$
    You prove it by showing that it has the properties of a homomorphism. How is a group homomorphism defined in your textbook/notes?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:31






  • 2




    $begingroup$
    The difficulty we have is that for anyone who has understood what is going on here the answer is essentially trivial, so it is hard to know what is tripping you up. How familiar are you with the symmetric groups? And why is it difficult to apply the definition you have of a homomorphism?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:43
















0












$begingroup$


Prove that $varphi: S_n to S_{n+1 }$, where for $sigmain S_{n}$: begin{pmatrix} 1 & 2 & dots & n \ 1sigma & 2sigma & dots & nsigma\ end{pmatrix}



$varphi(sigma)in S_{n+1}= begin{pmatrix} 1 & 2 & dots & n & n+1 \ 1sigma & 2sigma & dots & nsigma & (n+1)sigma \ end{pmatrix} $
is a homomorphism.



Wondering if anyone can explain how to prove this. Thanks.




From the comments:




I literally have no idea where to start, hence why I haven't given details of where ive got stuck as I have no idea what methods I should use to even start it. I have done a few questions similar using matrices, trigonometric functions and other functions and found them fine. I have been using the standard definition for homomorphism that is that φ(ab)=φ(a)φ(b). However, with this question this did not seem to get me very far at all and I was wondering if there was something more clever or a silly detail I was overlooking.







permutations group-homomorphism







share|cite|improve this question











$endgroup$












  • $begingroup$
    How have you proved this?
    $endgroup$
    – Finn Johnson
    Dec 31 '18 at 15:23






  • 1




    $begingroup$
    Welcome to MSE @FinnJohnson. Your question may attract some negative votes. I let you here why it can be like that: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc."
    $endgroup$
    – idriskameni
    Dec 31 '18 at 15:30








  • 2




    $begingroup$
    If you want answers, at the very least don't post questions as if you were instructing us to do what you want.
    $endgroup$
    – José Carlos Santos
    Dec 31 '18 at 15:30






  • 2




    $begingroup$
    You prove it by showing that it has the properties of a homomorphism. How is a group homomorphism defined in your textbook/notes?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:31






  • 2




    $begingroup$
    The difficulty we have is that for anyone who has understood what is going on here the answer is essentially trivial, so it is hard to know what is tripping you up. How familiar are you with the symmetric groups? And why is it difficult to apply the definition you have of a homomorphism?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:43














0












0








0





$begingroup$


Prove that $varphi: S_n to S_{n+1 }$, where for $sigmain S_{n}$: begin{pmatrix} 1 & 2 & dots & n \ 1sigma & 2sigma & dots & nsigma\ end{pmatrix}



$varphi(sigma)in S_{n+1}= begin{pmatrix} 1 & 2 & dots & n & n+1 \ 1sigma & 2sigma & dots & nsigma & (n+1)sigma \ end{pmatrix} $
is a homomorphism.



Wondering if anyone can explain how to prove this. Thanks.




From the comments:




I literally have no idea where to start, hence why I haven't given details of where ive got stuck as I have no idea what methods I should use to even start it. I have done a few questions similar using matrices, trigonometric functions and other functions and found them fine. I have been using the standard definition for homomorphism that is that φ(ab)=φ(a)φ(b). However, with this question this did not seem to get me very far at all and I was wondering if there was something more clever or a silly detail I was overlooking.







permutations group-homomorphism







share|cite|improve this question











$endgroup$




Prove that $varphi: S_n to S_{n+1 }$, where for $sigmain S_{n}$: begin{pmatrix} 1 & 2 & dots & n \ 1sigma & 2sigma & dots & nsigma\ end{pmatrix}



$varphi(sigma)in S_{n+1}= begin{pmatrix} 1 & 2 & dots & n & n+1 \ 1sigma & 2sigma & dots & nsigma & (n+1)sigma \ end{pmatrix} $
is a homomorphism.



Wondering if anyone can explain how to prove this. Thanks.




From the comments:




I literally have no idea where to start, hence why I haven't given details of where ive got stuck as I have no idea what methods I should use to even start it. I have done a few questions similar using matrices, trigonometric functions and other functions and found them fine. I have been using the standard definition for homomorphism that is that φ(ab)=φ(a)φ(b). However, with this question this did not seem to get me very far at all and I was wondering if there was something more clever or a silly detail I was overlooking.







permutations group-homomorphism




permutations group-homomorphism






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 31 '18 at 18:26









Shaun

9,310113684




9,310113684










asked Dec 31 '18 at 15:20









Finn JohnsonFinn Johnson

211




211












  • $begingroup$
    How have you proved this?
    $endgroup$
    – Finn Johnson
    Dec 31 '18 at 15:23






  • 1




    $begingroup$
    Welcome to MSE @FinnJohnson. Your question may attract some negative votes. I let you here why it can be like that: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc."
    $endgroup$
    – idriskameni
    Dec 31 '18 at 15:30








  • 2




    $begingroup$
    If you want answers, at the very least don't post questions as if you were instructing us to do what you want.
    $endgroup$
    – José Carlos Santos
    Dec 31 '18 at 15:30






  • 2




    $begingroup$
    You prove it by showing that it has the properties of a homomorphism. How is a group homomorphism defined in your textbook/notes?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:31






  • 2




    $begingroup$
    The difficulty we have is that for anyone who has understood what is going on here the answer is essentially trivial, so it is hard to know what is tripping you up. How familiar are you with the symmetric groups? And why is it difficult to apply the definition you have of a homomorphism?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:43


















  • $begingroup$
    How have you proved this?
    $endgroup$
    – Finn Johnson
    Dec 31 '18 at 15:23






  • 1




    $begingroup$
    Welcome to MSE @FinnJohnson. Your question may attract some negative votes. I let you here why it can be like that: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc."
    $endgroup$
    – idriskameni
    Dec 31 '18 at 15:30








  • 2




    $begingroup$
    If you want answers, at the very least don't post questions as if you were instructing us to do what you want.
    $endgroup$
    – José Carlos Santos
    Dec 31 '18 at 15:30






  • 2




    $begingroup$
    You prove it by showing that it has the properties of a homomorphism. How is a group homomorphism defined in your textbook/notes?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:31






  • 2




    $begingroup$
    The difficulty we have is that for anyone who has understood what is going on here the answer is essentially trivial, so it is hard to know what is tripping you up. How familiar are you with the symmetric groups? And why is it difficult to apply the definition you have of a homomorphism?
    $endgroup$
    – Mark Bennet
    Dec 31 '18 at 15:43
















$begingroup$
How have you proved this?
$endgroup$
– Finn Johnson
Dec 31 '18 at 15:23




$begingroup$
How have you proved this?
$endgroup$
– Finn Johnson
Dec 31 '18 at 15:23




1




1




$begingroup$
Welcome to MSE @FinnJohnson. Your question may attract some negative votes. I let you here why it can be like that: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc."
$endgroup$
– idriskameni
Dec 31 '18 at 15:30






$begingroup$
Welcome to MSE @FinnJohnson. Your question may attract some negative votes. I let you here why it can be like that: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc."
$endgroup$
– idriskameni
Dec 31 '18 at 15:30






2




2




$begingroup$
If you want answers, at the very least don't post questions as if you were instructing us to do what you want.
$endgroup$
– José Carlos Santos
Dec 31 '18 at 15:30




$begingroup$
If you want answers, at the very least don't post questions as if you were instructing us to do what you want.
$endgroup$
– José Carlos Santos
Dec 31 '18 at 15:30




2




2




$begingroup$
You prove it by showing that it has the properties of a homomorphism. How is a group homomorphism defined in your textbook/notes?
$endgroup$
– Mark Bennet
Dec 31 '18 at 15:31




$begingroup$
You prove it by showing that it has the properties of a homomorphism. How is a group homomorphism defined in your textbook/notes?
$endgroup$
– Mark Bennet
Dec 31 '18 at 15:31




2




2




$begingroup$
The difficulty we have is that for anyone who has understood what is going on here the answer is essentially trivial, so it is hard to know what is tripping you up. How familiar are you with the symmetric groups? And why is it difficult to apply the definition you have of a homomorphism?
$endgroup$
– Mark Bennet
Dec 31 '18 at 15:43




$begingroup$
The difficulty we have is that for anyone who has understood what is going on here the answer is essentially trivial, so it is hard to know what is tripping you up. How familiar are you with the symmetric groups? And why is it difficult to apply the definition you have of a homomorphism?
$endgroup$
– Mark Bennet
Dec 31 '18 at 15:43










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