Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$. [closed]
Let $G$ be a finite group, $Nmathrel{lhd}G$ a normal subgroup of $G$, and $Hleq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $gcd(|H|,|N|)=1$). Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.
I am trying to show that there is an injective homomorphism $varphi:Hrightarrow G/N$ , but I have no clue for the next step.
group-theory finite-groups normal-subgroups group-isomorphism quotient-group
edited Dec 8 at 23:13
Batominovski
33.7k33292
asked Dec 8 at 22:29
aaron wang
212
closed as unclear what you're asking by Shaun, user10354138, Cesareo, Rebellos, GNUSupporter 8964民主女神 地下教會 Dec 9 at 14:05
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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Let $G$ be a finite group, $Nmathrel{lhd}G$ a normal subgroup of $G$, and $Hleq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $gcd(|H|,|N|)=1$). Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.
I am trying to show that there is an injective homomorphism $varphi:Hrightarrow G/N$ , but I have no clue for the next step.
group-theory finite-groups normal-subgroups group-isomorphism quotient-group
edited Dec 8 at 23:13
Batominovski
33.7k33292
asked Dec 8 at 22:29
aaron wang
212
closed as unclear what you're asking by Shaun, user10354138, Cesareo, Rebellos, GNUSupporter 8964民主女神 地下教會 Dec 9 at 14:05
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
Do You mean $N triangleleft G$ instead of $H triangleleft G$? And also $G/N$ instead of $G/H$? as $G/H$ may not be a group.
– mathnoob
Dec 8 at 22:30
Hint. Think about what the natural homomorphism from $G$ to $G/N$ does to $H$. Can it send anything other than the identity to the identity?
– Ethan Bolker
Dec 8 at 23:03
1
The second isomorphism theorem says $HN/N cong H/(H cap N)$. Here $H cap N$ is trivial as $(|H|,|N|)=1$. So then $HN/N cong H$.
– mathnoob
Dec 8 at 23:16
add a comment |
Let $G$ be a finite group, $Nmathrel{lhd}G$ a normal subgroup of $G$, and $Hleq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $gcd(|H|,|N|)=1$). Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.
I am trying to show that there is an injective homomorphism $varphi:Hrightarrow G/N$ , but I have no clue for the next step.
group-theory finite-groups normal-subgroups group-isomorphism quotient-group
edited Dec 8 at 23:13
Batominovski
33.7k33292
asked Dec 8 at 22:29
aaron wang
212
Let $G$ be a finite group, $Nmathrel{lhd}G$ a normal subgroup of $G$, and $Hleq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $gcd(|H|,|N|)=1$). Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.
I am trying to show that there is an injective homomorphism $varphi:Hrightarrow G/N$ , but I have no clue for the next step.
group-theory finite-groups normal-subgroups group-isomorphism quotient-group
group-theory finite-groups normal-subgroups group-isomorphism quotient-group
edited Dec 8 at 23:13
Batominovski
33.7k33292
asked Dec 8 at 22:29
aaron wang
212
edited Dec 8 at 23:13
Batominovski
33.7k33292
asked Dec 8 at 22:29
aaron wang
212
edited Dec 8 at 23:13
Batominovski
33.7k33292
edited Dec 8 at 23:13
Batominovski
33.7k33292
edited Dec 8 at 23:13
Batominovski
33.7k33292
33.7k33292
asked Dec 8 at 22:29
aaron wang
212
asked Dec 8 at 22:29
aaron wang
212
asked Dec 8 at 22:29
aaron wang
212
212
closed as unclear what you're asking by Shaun, user10354138, Cesareo, Rebellos, GNUSupporter 8964民主女神 地下教會 Dec 9 at 14:05
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Shaun, user10354138, Cesareo, Rebellos, GNUSupporter 8964民主女神 地下教會 Dec 9 at 14:05
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
Do You mean $N triangleleft G$ instead of $H triangleleft G$? And also $G/N$ instead of $G/H$? as $G/H$ may not be a group.
– mathnoob
Dec 8 at 22:30
Hint. Think about what the natural homomorphism from $G$ to $G/N$ does to $H$. Can it send anything other than the identity to the identity?
– Ethan Bolker
Dec 8 at 23:03
1
The second isomorphism theorem says $HN/N cong H/(H cap N)$. Here $H cap N$ is trivial as $(|H|,|N|)=1$. So then $HN/N cong H$.
– mathnoob
Dec 8 at 23:16
add a comment |
4
Do You mean $N triangleleft G$ instead of $H triangleleft G$? And also $G/N$ instead of $G/H$? as $G/H$ may not be a group.
– mathnoob
Dec 8 at 22:30
Hint. Think about what the natural homomorphism from $G$ to $G/N$ does to $H$. Can it send anything other than the identity to the identity?
– Ethan Bolker
Dec 8 at 23:03
1
The second isomorphism theorem says $HN/N cong H/(H cap N)$. Here $H cap N$ is trivial as $(|H|,|N|)=1$. So then $HN/N cong H$.
– mathnoob
Dec 8 at 23:16
4
4
Do You mean $N triangleleft G$ instead of $H triangleleft G$? And also $G/N$ instead of $G/H$? as $G/H$ may not be a group.
– mathnoob
Dec 8 at 22:30
Do You mean $N triangleleft G$ instead of $H triangleleft G$? And also $G/N$ instead of $G/H$? as $G/H$ may not be a group.
– mathnoob
Dec 8 at 22:30
Hint. Think about what the natural homomorphism from $G$ to $G/N$ does to $H$. Can it send anything other than the identity to the identity?
– Ethan Bolker
Dec 8 at 23:03
Hint. Think about what the natural homomorphism from $G$ to $G/N$ does to $H$. Can it send anything other than the identity to the identity?
– Ethan Bolker
Dec 8 at 23:03
1
1
The second isomorphism theorem says $HN/N cong H/(H cap N)$. Here $H cap N$ is trivial as $(|H|,|N|)=1$. So then $HN/N cong H$.
– mathnoob
Dec 8 at 23:16
The second isomorphism theorem says $HN/N cong H/(H cap N)$. Here $H cap N$ is trivial as $(|H|,|N|)=1$. So then $HN/N cong H$.
– mathnoob
Dec 8 at 23:16
add a comment |
1 Answer
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Hint: Show that $HN$ is a subgroup of $G$ containing $N$. Prove that, under the canonical projection $Gto (G/N)$, the subgroup $HN$ of $G$ is mapped onto a subgroup of $G/N$ isomorphic to $H$ (this subgroup is clearly, $HN/N$). More generally, we have $HN/Ncong H/(Hcap N)$.
answered Dec 8 at 23:11
Batominovski
33.7k33292
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint: Show that $HN$ is a subgroup of $G$ containing $N$. Prove that, under the canonical projection $Gto (G/N)$, the subgroup $HN$ of $G$ is mapped onto a subgroup of $G/N$ isomorphic to $H$ (this subgroup is clearly, $HN/N$). More generally, we have $HN/Ncong H/(Hcap N)$.
answered Dec 8 at 23:11
Batominovski
33.7k33292
add a comment |
Hint: Show that $HN$ is a subgroup of $G$ containing $N$. Prove that, under the canonical projection $Gto (G/N)$, the subgroup $HN$ of $G$ is mapped onto a subgroup of $G/N$ isomorphic to $H$ (this subgroup is clearly, $HN/N$). More generally, we have $HN/Ncong H/(Hcap N)$.
answered Dec 8 at 23:11
Batominovski
33.7k33292
add a comment |
Hint: Show that $HN$ is a subgroup of $G$ containing $N$. Prove that, under the canonical projection $Gto (G/N)$, the subgroup $HN$ of $G$ is mapped onto a subgroup of $G/N$ isomorphic to $H$ (this subgroup is clearly, $HN/N$). More generally, we have $HN/Ncong H/(Hcap N)$.
answered Dec 8 at 23:11
Batominovski
33.7k33292
Hint: Show that $HN$ is a subgroup of $G$ containing $N$. Prove that, under the canonical projection $Gto (G/N)$, the subgroup $HN$ of $G$ is mapped onto a subgroup of $G/N$ isomorphic to $H$ (this subgroup is clearly, $HN/N$). More generally, we have $HN/Ncong H/(Hcap N)$.
answered Dec 8 at 23:11
Batominovski
33.7k33292
answered Dec 8 at 23:11
Batominovski
33.7k33292
answered Dec 8 at 23:11
Batominovski
33.7k33292
answered Dec 8 at 23:11
Batominovski
33.7k33292
33.7k33292
add a comment |
add a comment |
4
Do You mean $N triangleleft G$ instead of $H triangleleft G$? And also $G/N$ instead of $G/H$? as $G/H$ may not be a group.
– mathnoob
Dec 8 at 22:30
Hint. Think about what the natural homomorphism from $G$ to $G/N$ does to $H$. Can it send anything other than the identity to the identity?
– Ethan Bolker
Dec 8 at 23:03
1
The second isomorphism theorem says $HN/N cong H/(H cap N)$. Here $H cap N$ is trivial as $(|H|,|N|)=1$. So then $HN/N cong H$.
– mathnoob
Dec 8 at 23:16