Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H times G/K$ has a subgroup that is isomorphic...
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Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.
Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H times G/K$
So far I have that by the second isomorphism theorem, $H/(H ∩ K)$ is isomorphic to $HK/K$. I'm not sure where to go from here.
abstract-algebra normal-subgroups group-isomorphism group-homomorphism quotient-group
edited Nov 23 at 23:45
egreg
175k1383198
asked Nov 23 at 23:15
kmediate
115
add a comment |
up vote
1
down vote
favorite
Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.
Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H times G/K$
So far I have that by the second isomorphism theorem, $H/(H ∩ K)$ is isomorphic to $HK/K$. I'm not sure where to go from here.
abstract-algebra normal-subgroups group-isomorphism group-homomorphism quotient-group
edited Nov 23 at 23:45
egreg
175k1383198
asked Nov 23 at 23:15
kmediate
115
1
You should review the proofs of the isomorphism theorems. They will give you the insight you need to solve these problems.
– John Douma
Nov 23 at 23:25
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.
Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H times G/K$
So far I have that by the second isomorphism theorem, $H/(H ∩ K)$ is isomorphic to $HK/K$. I'm not sure where to go from here.
abstract-algebra normal-subgroups group-isomorphism group-homomorphism quotient-group
edited Nov 23 at 23:45
egreg
175k1383198
asked Nov 23 at 23:15
kmediate
115
Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.
Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H times G/K$
So far I have that by the second isomorphism theorem, $H/(H ∩ K)$ is isomorphic to $HK/K$. I'm not sure where to go from here.
abstract-algebra normal-subgroups group-isomorphism group-homomorphism quotient-group
abstract-algebra normal-subgroups group-isomorphism group-homomorphism quotient-group
edited Nov 23 at 23:45
egreg
175k1383198
asked Nov 23 at 23:15
kmediate
115
edited Nov 23 at 23:45
egreg
175k1383198
asked Nov 23 at 23:15
kmediate
115
edited Nov 23 at 23:45
egreg
175k1383198
edited Nov 23 at 23:45
egreg
175k1383198
edited Nov 23 at 23:45
egreg
175k1383198
175k1383198
asked Nov 23 at 23:15
kmediate
115
asked Nov 23 at 23:15
kmediate
115
asked Nov 23 at 23:15
kmediate
115
115
1
You should review the proofs of the isomorphism theorems. They will give you the insight you need to solve these problems.
– John Douma
Nov 23 at 23:25
add a comment |
1
You should review the proofs of the isomorphism theorems. They will give you the insight you need to solve these problems.
– John Douma
Nov 23 at 23:25
1
1
You should review the proofs of the isomorphism theorems. They will give you the insight you need to solve these problems.
– John Douma
Nov 23 at 23:25
You should review the proofs of the isomorphism theorems. They will give you the insight you need to solve these problems.
– John Douma
Nov 23 at 23:25
add a comment |
2 Answers
2
active
oldest
votes
up vote
2
down vote
Hint: there is a natural homomorphism
$$
varphicolon Gto G/Htimes G/K
$$
defined by $varphi(x)=(xH,xK)$. What's its kernel? What's its image in case $HK=G$?
answered Nov 23 at 23:49
egreg
175k1383198
add a comment |
up vote
2
down vote
Define
$$phi: Gto G/Htimes G/K;,;;phi x:=(xH,,xK)$$
Prove the above is a group homomorphism and show $;kerphi=Hcap K;$ . Finally, use the first isomorphism theorem.
If you understand the above then the second part should be easier.
answered Nov 23 at 23:49
DonAntonio
176k1491224
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Hint: there is a natural homomorphism
$$
varphicolon Gto G/Htimes G/K
$$
defined by $varphi(x)=(xH,xK)$. What's its kernel? What's its image in case $HK=G$?
answered Nov 23 at 23:49
egreg
175k1383198
add a comment |
up vote
2
down vote
Hint: there is a natural homomorphism
$$
varphicolon Gto G/Htimes G/K
$$
defined by $varphi(x)=(xH,xK)$. What's its kernel? What's its image in case $HK=G$?
answered Nov 23 at 23:49
egreg
175k1383198
add a comment |
up vote
2
down vote
up vote
2
down vote
Hint: there is a natural homomorphism
$$
varphicolon Gto G/Htimes G/K
$$
defined by $varphi(x)=(xH,xK)$. What's its kernel? What's its image in case $HK=G$?
answered Nov 23 at 23:49
egreg
175k1383198
Hint: there is a natural homomorphism
$$
varphicolon Gto G/Htimes G/K
$$
defined by $varphi(x)=(xH,xK)$. What's its kernel? What's its image in case $HK=G$?
answered Nov 23 at 23:49
egreg
175k1383198
answered Nov 23 at 23:49
egreg
175k1383198
answered Nov 23 at 23:49
egreg
175k1383198
answered Nov 23 at 23:49
egreg
175k1383198
175k1383198
add a comment |
add a comment |
up vote
2
down vote
Define
$$phi: Gto G/Htimes G/K;,;;phi x:=(xH,,xK)$$
Prove the above is a group homomorphism and show $;kerphi=Hcap K;$ . Finally, use the first isomorphism theorem.
If you understand the above then the second part should be easier.
answered Nov 23 at 23:49
DonAntonio
176k1491224
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
add a comment |
up vote
2
down vote
Define
$$phi: Gto G/Htimes G/K;,;;phi x:=(xH,,xK)$$
Prove the above is a group homomorphism and show $;kerphi=Hcap K;$ . Finally, use the first isomorphism theorem.
If you understand the above then the second part should be easier.
answered Nov 23 at 23:49
DonAntonio
176k1491224
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
add a comment |
up vote
2
down vote
up vote
2
down vote
Define
$$phi: Gto G/Htimes G/K;,;;phi x:=(xH,,xK)$$
Prove the above is a group homomorphism and show $;kerphi=Hcap K;$ . Finally, use the first isomorphism theorem.
If you understand the above then the second part should be easier.
answered Nov 23 at 23:49
DonAntonio
176k1491224
Define
$$phi: Gto G/Htimes G/K;,;;phi x:=(xH,,xK)$$
Prove the above is a group homomorphism and show $;kerphi=Hcap K;$ . Finally, use the first isomorphism theorem.
If you understand the above then the second part should be easier.
answered Nov 23 at 23:49
DonAntonio
176k1491224
answered Nov 23 at 23:49
DonAntonio
176k1491224
answered Nov 23 at 23:49
DonAntonio
176k1491224
answered Nov 23 at 23:49
DonAntonio
176k1491224
176k1491224
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
add a comment |
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
For the second part, I started with: define φ: HK →G/H×G/K by φ(hk)=(xH, xK). I then went through with proving it was a homomorphism... am I on the right track to proving G/(H∩K) is isomorphic to G/H×G/K? or am I missing an easier way to do it with the isomorphism theorems?
– kmediate
Nov 24 at 4:00
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
@KatieMediate Use the very same map for the 2nd part, and use that $;G=HK;$ ... BUT you almost should use that $;HK=KH;$ since this is a sufficient and necessary condition for the product of two subgroups to be a subgroup again. With this you shouldn't have tough problems to show the map now is surjective.
– DonAntonio
Nov 24 at 11:38
add a comment |
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You should review the proofs of the isomorphism theorems. They will give you the insight you need to solve these problems.
– John Douma
Nov 23 at 23:25