Mixing
If there exists an element $x$ of order $n$ in $G'$, then we must have an element of order $n$ in $G$?
$begingroup$
If there exists a homomorphism from $G$ to $G'$, if there exists an element $x$ of order $n$ in $G'$, then we must have an element of order $n$ in $G$
What if the homomorphism is onto?
Counter example of the first statement : Take the trivial homomorphism from $Z_2 $ to $Z_8$
I'll try to prove the onto case :
Let $xin G', ; |x|=n Rightarrow exists yin G ; text{s.t} ; phi(y)=x Rightarrow |x| ; text{divides} ; |y| Rightarrow |y|=nk ; text{f.s} ; k in N Rightarrow |y^{k}| = n $ and so $y^k$ is required element of $G$
Is this correct?
group-theory proof-verification group-homomorphism
asked Jan 25 at 4:34
AbhayAbhay
3789
$endgroup$
add a comment |
$begingroup$
If there exists a homomorphism from $G$ to $G'$, if there exists an element $x$ of order $n$ in $G'$, then we must have an element of order $n$ in $G$
What if the homomorphism is onto?
Counter example of the first statement : Take the trivial homomorphism from $Z_2 $ to $Z_8$
I'll try to prove the onto case :
Let $xin G', ; |x|=n Rightarrow exists yin G ; text{s.t} ; phi(y)=x Rightarrow |x| ; text{divides} ; |y| Rightarrow |y|=nk ; text{f.s} ; k in N Rightarrow |y^{k}| = n $ and so $y^k$ is required element of $G$
Is this correct?
group-theory proof-verification group-homomorphism
asked Jan 25 at 4:34
AbhayAbhay
3789
$endgroup$
3
$begingroup$
Are you saying statement of first paragraph is true, or asking if it is? And by second question are you just asking about that statement under the assumption the homomorphism is onto?
$endgroup$
– coffeemath
Jan 25 at 4:39
2
$begingroup$
This question is ill-posed at present. Are you hoping the community will help you prove the first sentence? Please make it clear, and demonstrate how you have attempted to solve the problem - this is not a forum where we do your homework for you.
$endgroup$
– bounceback
Jan 25 at 4:39
1
$begingroup$
@coffeemath can you see the edited post please
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@bounceback please see the edited post now
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
Well you've butchered the implies symbol, and I don't like your 'f.s', but the argument appears sound. You should be fine.
$endgroup$
– bounceback
Jan 25 at 18:00
add a comment |
$begingroup$
If there exists a homomorphism from $G$ to $G'$, if there exists an element $x$ of order $n$ in $G'$, then we must have an element of order $n$ in $G$
What if the homomorphism is onto?
Counter example of the first statement : Take the trivial homomorphism from $Z_2 $ to $Z_8$
I'll try to prove the onto case :
Let $xin G', ; |x|=n Rightarrow exists yin G ; text{s.t} ; phi(y)=x Rightarrow |x| ; text{divides} ; |y| Rightarrow |y|=nk ; text{f.s} ; k in N Rightarrow |y^{k}| = n $ and so $y^k$ is required element of $G$
Is this correct?
group-theory proof-verification group-homomorphism
asked Jan 25 at 4:34
AbhayAbhay
3789
$endgroup$
If there exists a homomorphism from $G$ to $G'$, if there exists an element $x$ of order $n$ in $G'$, then we must have an element of order $n$ in $G$
What if the homomorphism is onto?
Counter example of the first statement : Take the trivial homomorphism from $Z_2 $ to $Z_8$
I'll try to prove the onto case :
Let $xin G', ; |x|=n Rightarrow exists yin G ; text{s.t} ; phi(y)=x Rightarrow |x| ; text{divides} ; |y| Rightarrow |y|=nk ; text{f.s} ; k in N Rightarrow |y^{k}| = n $ and so $y^k$ is required element of $G$
Is this correct?
group-theory proof-verification group-homomorphism
group-theory proof-verification group-homomorphism
asked Jan 25 at 4:34
AbhayAbhay
3789
asked Jan 25 at 4:34
AbhayAbhay
3789
edited Jan 25 at 5:01
Abhay
asked Jan 25 at 4:34
AbhayAbhay
3789
asked Jan 25 at 4:34
AbhayAbhay
3789
asked Jan 25 at 4:34
AbhayAbhay
3789
3789
3
$begingroup$
Are you saying statement of first paragraph is true, or asking if it is? And by second question are you just asking about that statement under the assumption the homomorphism is onto?
$endgroup$
– coffeemath
Jan 25 at 4:39
2
$begingroup$
This question is ill-posed at present. Are you hoping the community will help you prove the first sentence? Please make it clear, and demonstrate how you have attempted to solve the problem - this is not a forum where we do your homework for you.
$endgroup$
– bounceback
Jan 25 at 4:39
1
$begingroup$
@coffeemath can you see the edited post please
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@bounceback please see the edited post now
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
Well you've butchered the implies symbol, and I don't like your 'f.s', but the argument appears sound. You should be fine.
$endgroup$
– bounceback
Jan 25 at 18:00
add a comment |
3
$begingroup$
Are you saying statement of first paragraph is true, or asking if it is? And by second question are you just asking about that statement under the assumption the homomorphism is onto?
$endgroup$
– coffeemath
Jan 25 at 4:39
2
$begingroup$
This question is ill-posed at present. Are you hoping the community will help you prove the first sentence? Please make it clear, and demonstrate how you have attempted to solve the problem - this is not a forum where we do your homework for you.
$endgroup$
– bounceback
Jan 25 at 4:39
1
$begingroup$
@coffeemath can you see the edited post please
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@bounceback please see the edited post now
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
Well you've butchered the implies symbol, and I don't like your 'f.s', but the argument appears sound. You should be fine.
$endgroup$
– bounceback
Jan 25 at 18:00
3
3
$begingroup$
Are you saying statement of first paragraph is true, or asking if it is? And by second question are you just asking about that statement under the assumption the homomorphism is onto?
$endgroup$
– coffeemath
Jan 25 at 4:39
$begingroup$
Are you saying statement of first paragraph is true, or asking if it is? And by second question are you just asking about that statement under the assumption the homomorphism is onto?
$endgroup$
– coffeemath
Jan 25 at 4:39
2
2
$begingroup$
This question is ill-posed at present. Are you hoping the community will help you prove the first sentence? Please make it clear, and demonstrate how you have attempted to solve the problem - this is not a forum where we do your homework for you.
$endgroup$
– bounceback
Jan 25 at 4:39
$begingroup$
This question is ill-posed at present. Are you hoping the community will help you prove the first sentence? Please make it clear, and demonstrate how you have attempted to solve the problem - this is not a forum where we do your homework for you.
$endgroup$
– bounceback
Jan 25 at 4:39
1
1
$begingroup$
@coffeemath can you see the edited post please
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@coffeemath can you see the edited post please
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@bounceback please see the edited post now
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@bounceback please see the edited post now
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
Well you've butchered the implies symbol, and I don't like your 'f.s', but the argument appears sound. You should be fine.
$endgroup$
– bounceback
Jan 25 at 18:00
$begingroup$
Well you've butchered the implies symbol, and I don't like your 'f.s', but the argument appears sound. You should be fine.
$endgroup$
– bounceback
Jan 25 at 18:00
add a comment |
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$begingroup$
Are you saying statement of first paragraph is true, or asking if it is? And by second question are you just asking about that statement under the assumption the homomorphism is onto?
$endgroup$
– coffeemath
Jan 25 at 4:39
2
$begingroup$
This question is ill-posed at present. Are you hoping the community will help you prove the first sentence? Please make it clear, and demonstrate how you have attempted to solve the problem - this is not a forum where we do your homework for you.
$endgroup$
– bounceback
Jan 25 at 4:39
1
$begingroup$
@coffeemath can you see the edited post please
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
@bounceback please see the edited post now
$endgroup$
– Abhay
Jan 25 at 5:01
$begingroup$
Well you've butchered the implies symbol, and I don't like your 'f.s', but the argument appears sound. You should be fine.
$endgroup$
– bounceback
Jan 25 at 18:00