The order of an element in a quotient group












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Suppose $G$ is a finite group, that $H$ is a subgroup of $G$, and that $N$ is a normal subgroup of $G$. Suppose that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime.
Consider the quotient group $G/N$ and let $h in H$. Determine the order of the element $hN$ in the group $G/N$.



Attempt:



Then $|G/N|=m$ , $|H|=n$ , $operatorname{gcd}(m,n)=1$



Order of the element $hN$ in $G/N$ is $k$ where $k$ is the smallest positive integer such that $h^k$ belongs to $N$. $(hN)^k = (h^k)(N)=1 $ iff $h^k$ belongs to $N$.



I am not sure how to use $|H|=n$ and $gcd(m,n)=1$ to find the order of $hN$.










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  • The order of $hN$ divides the order of $h$.
    – Tortoise
    Nov 19 '18 at 10:16
















1














Suppose $G$ is a finite group, that $H$ is a subgroup of $G$, and that $N$ is a normal subgroup of $G$. Suppose that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime.
Consider the quotient group $G/N$ and let $h in H$. Determine the order of the element $hN$ in the group $G/N$.



Attempt:



Then $|G/N|=m$ , $|H|=n$ , $operatorname{gcd}(m,n)=1$



Order of the element $hN$ in $G/N$ is $k$ where $k$ is the smallest positive integer such that $h^k$ belongs to $N$. $(hN)^k = (h^k)(N)=1 $ iff $h^k$ belongs to $N$.



I am not sure how to use $|H|=n$ and $gcd(m,n)=1$ to find the order of $hN$.










share|cite|improve this question
























  • The order of $hN$ divides the order of $h$.
    – Tortoise
    Nov 19 '18 at 10:16














1












1








1







Suppose $G$ is a finite group, that $H$ is a subgroup of $G$, and that $N$ is a normal subgroup of $G$. Suppose that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime.
Consider the quotient group $G/N$ and let $h in H$. Determine the order of the element $hN$ in the group $G/N$.



Attempt:



Then $|G/N|=m$ , $|H|=n$ , $operatorname{gcd}(m,n)=1$



Order of the element $hN$ in $G/N$ is $k$ where $k$ is the smallest positive integer such that $h^k$ belongs to $N$. $(hN)^k = (h^k)(N)=1 $ iff $h^k$ belongs to $N$.



I am not sure how to use $|H|=n$ and $gcd(m,n)=1$ to find the order of $hN$.










share|cite|improve this question















Suppose $G$ is a finite group, that $H$ is a subgroup of $G$, and that $N$ is a normal subgroup of $G$. Suppose that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime.
Consider the quotient group $G/N$ and let $h in H$. Determine the order of the element $hN$ in the group $G/N$.



Attempt:



Then $|G/N|=m$ , $|H|=n$ , $operatorname{gcd}(m,n)=1$



Order of the element $hN$ in $G/N$ is $k$ where $k$ is the smallest positive integer such that $h^k$ belongs to $N$. $(hN)^k = (h^k)(N)=1 $ iff $h^k$ belongs to $N$.



I am not sure how to use $|H|=n$ and $gcd(m,n)=1$ to find the order of $hN$.







abstract-algebra group-theory normal-subgroups quotient-group






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edited Nov 19 '18 at 0:34

























asked Nov 19 '18 at 0:26









J. Dawson

143




143












  • The order of $hN$ divides the order of $h$.
    – Tortoise
    Nov 19 '18 at 10:16


















  • The order of $hN$ divides the order of $h$.
    – Tortoise
    Nov 19 '18 at 10:16
















The order of $hN$ divides the order of $h$.
– Tortoise
Nov 19 '18 at 10:16




The order of $hN$ divides the order of $h$.
– Tortoise
Nov 19 '18 at 10:16










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I got the idea now. $(|H|,|G/N|)=1$ is key to do this problem. Notice that $|hN|||h|$ (check) and also $|h|||H|$ so that $|hN|||H|$ but $|hN||(G/N)$ since $|H|$ and $|G/N|$ are relatively prime so that $|hN|=1$. Thus $hN=N$ and then $Hsubset N.$ This finishes the proof.






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    I got the idea now. $(|H|,|G/N|)=1$ is key to do this problem. Notice that $|hN|||h|$ (check) and also $|h|||H|$ so that $|hN|||H|$ but $|hN||(G/N)$ since $|H|$ and $|G/N|$ are relatively prime so that $|hN|=1$. Thus $hN=N$ and then $Hsubset N.$ This finishes the proof.






    share|cite|improve this answer




























      1














      I got the idea now. $(|H|,|G/N|)=1$ is key to do this problem. Notice that $|hN|||h|$ (check) and also $|h|||H|$ so that $|hN|||H|$ but $|hN||(G/N)$ since $|H|$ and $|G/N|$ are relatively prime so that $|hN|=1$. Thus $hN=N$ and then $Hsubset N.$ This finishes the proof.






      share|cite|improve this answer


























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        1






        I got the idea now. $(|H|,|G/N|)=1$ is key to do this problem. Notice that $|hN|||h|$ (check) and also $|h|||H|$ so that $|hN|||H|$ but $|hN||(G/N)$ since $|H|$ and $|G/N|$ are relatively prime so that $|hN|=1$. Thus $hN=N$ and then $Hsubset N.$ This finishes the proof.






        share|cite|improve this answer














        I got the idea now. $(|H|,|G/N|)=1$ is key to do this problem. Notice that $|hN|||h|$ (check) and also $|h|||H|$ so that $|hN|||H|$ but $|hN||(G/N)$ since $|H|$ and $|G/N|$ are relatively prime so that $|hN|=1$. Thus $hN=N$ and then $Hsubset N.$ This finishes the proof.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 19 '18 at 2:13

























        answered Nov 19 '18 at 0:46









        gb2017

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