Question concerning isomorphism of quotient groups
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I saw a video that tells that if $p, q$ are integers such that $p|q$, then $Z_q/Z_p$ is isomorphic to $Z_{q/p}$. If it is true, can you give me a hint about how can I prove this?
abstract-algebra quotient-group
asked Nov 24 '18 at 1:00
user573497user573497
16619
$endgroup$
add a comment |
$begingroup$
I saw a video that tells that if $p, q$ are integers such that $p|q$, then $Z_q/Z_p$ is isomorphic to $Z_{q/p}$. If it is true, can you give me a hint about how can I prove this?
abstract-algebra quotient-group
asked Nov 24 '18 at 1:00
user573497user573497
16619
$endgroup$
$begingroup$
Look at $<$p$>$, the cyclic subgroup generated by p in $mathbb{Z}_q$. How many elements are there? Since everything is abelian, the quotient group exists. How many elements are there in the quotient group?
$endgroup$
– Joel Pereira
Nov 24 '18 at 1:04
$begingroup$
What's $Z_q / Z_p$? I'm not just quibbling with notation. So I'm asking "What's $(mathbb{Z}/qmathbb{Z})/(mathbb{Z}/pmathbb{Z})$?" The collection $Z_p$ isn't a subset of $Z_q$. Perhaps $Z_q / pZ_q$?
$endgroup$
– Eric Towers
Nov 24 '18 at 1:05
add a comment |
$begingroup$
I saw a video that tells that if $p, q$ are integers such that $p|q$, then $Z_q/Z_p$ is isomorphic to $Z_{q/p}$. If it is true, can you give me a hint about how can I prove this?
abstract-algebra quotient-group
asked Nov 24 '18 at 1:00
user573497user573497
16619
$endgroup$
I saw a video that tells that if $p, q$ are integers such that $p|q$, then $Z_q/Z_p$ is isomorphic to $Z_{q/p}$. If it is true, can you give me a hint about how can I prove this?
abstract-algebra quotient-group
abstract-algebra quotient-group
asked Nov 24 '18 at 1:00
user573497user573497
16619
asked Nov 24 '18 at 1:00
user573497user573497
16619
asked Nov 24 '18 at 1:00
user573497user573497
16619
asked Nov 24 '18 at 1:00
user573497user573497
16619
asked Nov 24 '18 at 1:00
user573497user573497
16619
16619
$begingroup$
Look at $<$p$>$, the cyclic subgroup generated by p in $mathbb{Z}_q$. How many elements are there? Since everything is abelian, the quotient group exists. How many elements are there in the quotient group?
$endgroup$
– Joel Pereira
Nov 24 '18 at 1:04
$begingroup$
What's $Z_q / Z_p$? I'm not just quibbling with notation. So I'm asking "What's $(mathbb{Z}/qmathbb{Z})/(mathbb{Z}/pmathbb{Z})$?" The collection $Z_p$ isn't a subset of $Z_q$. Perhaps $Z_q / pZ_q$?
$endgroup$
– Eric Towers
Nov 24 '18 at 1:05
add a comment |
$begingroup$
Look at $<$p$>$, the cyclic subgroup generated by p in $mathbb{Z}_q$. How many elements are there? Since everything is abelian, the quotient group exists. How many elements are there in the quotient group?
$endgroup$
– Joel Pereira
Nov 24 '18 at 1:04
$begingroup$
What's $Z_q / Z_p$? I'm not just quibbling with notation. So I'm asking "What's $(mathbb{Z}/qmathbb{Z})/(mathbb{Z}/pmathbb{Z})$?" The collection $Z_p$ isn't a subset of $Z_q$. Perhaps $Z_q / pZ_q$?
$endgroup$
– Eric Towers
Nov 24 '18 at 1:05
$begingroup$
Look at $<$p$>$, the cyclic subgroup generated by p in $mathbb{Z}_q$. How many elements are there? Since everything is abelian, the quotient group exists. How many elements are there in the quotient group?
$endgroup$
– Joel Pereira
Nov 24 '18 at 1:04
$begingroup$
Look at $<$p$>$, the cyclic subgroup generated by p in $mathbb{Z}_q$. How many elements are there? Since everything is abelian, the quotient group exists. How many elements are there in the quotient group?
$endgroup$
– Joel Pereira
Nov 24 '18 at 1:04
$begingroup$
What's $Z_q / Z_p$? I'm not just quibbling with notation. So I'm asking "What's $(mathbb{Z}/qmathbb{Z})/(mathbb{Z}/pmathbb{Z})$?" The collection $Z_p$ isn't a subset of $Z_q$. Perhaps $Z_q / pZ_q$?
$endgroup$
– Eric Towers
Nov 24 '18 at 1:05
$begingroup$
What's $Z_q / Z_p$? I'm not just quibbling with notation. So I'm asking "What's $(mathbb{Z}/qmathbb{Z})/(mathbb{Z}/pmathbb{Z})$?" The collection $Z_p$ isn't a subset of $Z_q$. Perhaps $Z_q / pZ_q$?
$endgroup$
– Eric Towers
Nov 24 '18 at 1:05
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
Consider the homomorphism:
begin{align}
mathbf Z/qmathbf Z&longrightarrow mathbf Z/pmathbf Z\
nbmod q&longmapsto nbmod p
end{align}
Check it is well defined and surjective. What is its kernel?
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
Consider the homomorphism:
begin{align}
mathbf Z/qmathbf Z&longrightarrow mathbf Z/pmathbf Z\
nbmod q&longmapsto nbmod p
end{align}
Check it is well defined and surjective. What is its kernel?
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
$endgroup$
add a comment |
$begingroup$
Hint:
Consider the homomorphism:
begin{align}
mathbf Z/qmathbf Z&longrightarrow mathbf Z/pmathbf Z\
nbmod q&longmapsto nbmod p
end{align}
Check it is well defined and surjective. What is its kernel?
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
$endgroup$
add a comment |
$begingroup$
Hint:
Consider the homomorphism:
begin{align}
mathbf Z/qmathbf Z&longrightarrow mathbf Z/pmathbf Z\
nbmod q&longmapsto nbmod p
end{align}
Check it is well defined and surjective. What is its kernel?
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
$endgroup$
Hint:
Consider the homomorphism:
begin{align}
mathbf Z/qmathbf Z&longrightarrow mathbf Z/pmathbf Z\
nbmod q&longmapsto nbmod p
end{align}
Check it is well defined and surjective. What is its kernel?
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
answered Nov 24 '18 at 1:33
BernardBernard
119k639112
119k639112
add a comment |
add a comment |
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$begingroup$
Look at $<$p$>$, the cyclic subgroup generated by p in $mathbb{Z}_q$. How many elements are there? Since everything is abelian, the quotient group exists. How many elements are there in the quotient group?
$endgroup$
– Joel Pereira
Nov 24 '18 at 1:04
$begingroup$
What's $Z_q / Z_p$? I'm not just quibbling with notation. So I'm asking "What's $(mathbb{Z}/qmathbb{Z})/(mathbb{Z}/pmathbb{Z})$?" The collection $Z_p$ isn't a subset of $Z_q$. Perhaps $Z_q / pZ_q$?
$endgroup$
– Eric Towers
Nov 24 '18 at 1:05