Let $phi:Bbb Z_nrightarrow G$ s.t. $phi(i)=h^i$ for $0le ile n$. Give necessary and sufficient condition for...
$begingroup$
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
edited Dec 22 '18 at 11:37
Shaun
9,366113684
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
$endgroup$
|
show 4 more comments
$begingroup$
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
edited Dec 22 '18 at 11:37
Shaun
9,366113684
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
$endgroup$
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
|
show 4 more comments
$begingroup$
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
edited Dec 22 '18 at 11:37
Shaun
9,366113684
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
$endgroup$
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
edited Dec 22 '18 at 11:37
Shaun
9,366113684
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
edited Dec 22 '18 at 11:37
Shaun
9,366113684
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
edited Dec 22 '18 at 11:37
Shaun
9,366113684
edited Dec 22 '18 at 11:37
Shaun
9,366113684
edited Dec 22 '18 at 11:37
Shaun
9,366113684
9,366113684
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
asked Dec 21 '17 at 18:22
user2820579user2820579
771415
771415
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
|
show 4 more comments
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
|
show 4 more comments
1 Answer
1
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oldest
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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.
$endgroup$
add a comment |
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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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered Dec 22 '18 at 11:29
community wiki
Shaun
add a comment |
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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