Is every right coset a subgroup of the group? [closed]
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Is every right coset a subgroup of the group?
group-theory
edited Nov 28 '18 at 23:17
Shaun
8,832113681
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
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closed as off-topic by Shaun, KReiser, José Carlos Santos, amWhy, Shailesh Nov 29 '18 at 0:27
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, KReiser, José Carlos Santos, amWhy, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
Is every right coset a subgroup of the group?
group-theory
edited Nov 28 '18 at 23:17
Shaun
8,832113681
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
$endgroup$
closed as off-topic by Shaun, KReiser, José Carlos Santos, amWhy, Shailesh Nov 29 '18 at 0:27
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, KReiser, José Carlos Santos, amWhy, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
5
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No. In fact, only one of the cosets contains the identity element.
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– Darío G
Mar 18 '16 at 10:11
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$begingroup$
Is every right coset a subgroup of the group?
group-theory
edited Nov 28 '18 at 23:17
Shaun
8,832113681
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
$endgroup$
Is every right coset a subgroup of the group?
group-theory
group-theory
edited Nov 28 '18 at 23:17
Shaun
8,832113681
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
edited Nov 28 '18 at 23:17
Shaun
8,832113681
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
edited Nov 28 '18 at 23:17
Shaun
8,832113681
edited Nov 28 '18 at 23:17
Shaun
8,832113681
edited Nov 28 '18 at 23:17
Shaun
8,832113681
8,832113681
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
asked Mar 18 '16 at 10:10
Kiran PatelKiran Patel
41
41
closed as off-topic by Shaun, KReiser, José Carlos Santos, amWhy, Shailesh Nov 29 '18 at 0:27
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, KReiser, José Carlos Santos, amWhy, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Shaun, KReiser, José Carlos Santos, amWhy, Shailesh Nov 29 '18 at 0:27
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, KReiser, José Carlos Santos, amWhy, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
No. In fact, only one of the cosets contains the identity element.
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– Darío G
Mar 18 '16 at 10:11
add a comment |
5
$begingroup$
No. In fact, only one of the cosets contains the identity element.
$endgroup$
– Darío G
Mar 18 '16 at 10:11
5
5
$begingroup$
No. In fact, only one of the cosets contains the identity element.
$endgroup$
– Darío G
Mar 18 '16 at 10:11
$begingroup$
No. In fact, only one of the cosets contains the identity element.
$endgroup$
– Darío G
Mar 18 '16 at 10:11
add a comment |
3 Answers
3
active
oldest
votes
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To expand on the other answers with an example: Wikipedia has this neat diagram about cosets.
It portrays
- the group $G = mathbb Z_8$ under addition,
- its subgroup $H = {0, 4}$,
- all cosets of $H$ in $G$.
Clearly, only $H$ contains the identity element $0$, so only $H$ is a group. The cosets $1+H$ and $2+H$ and $3+H$ don’t contain $0$, so they fail to satisfy this important group property.
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
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add a comment |
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No, Let $(G,circ ,e)$ be our group, Note that for every $gin G$ and $Hle G$ s.t $gneq e$, We get that $enotin gH$ And therefor cannot be a subgroup of $H$ nor $G$ (the same goes for left cosets)
answered Mar 18 '16 at 10:16
shasha
987615
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add a comment |
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If $H leq G$ then $aHbH=abH quad forall a,bin G$ if and only if $H$ is a normal subgroup of $G$ i.e $aHa^{-1}=H quad forall ain G$ .
For a counter example $ langle (1,2)rangle leq S_3 $.
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I thought that he,she had asked that when cosets make a group :)
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– user217174
Mar 18 '16 at 14:20
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
To expand on the other answers with an example: Wikipedia has this neat diagram about cosets.
It portrays
- the group $G = mathbb Z_8$ under addition,
- its subgroup $H = {0, 4}$,
- all cosets of $H$ in $G$.
Clearly, only $H$ contains the identity element $0$, so only $H$ is a group. The cosets $1+H$ and $2+H$ and $3+H$ don’t contain $0$, so they fail to satisfy this important group property.
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
$endgroup$
add a comment |
$begingroup$
To expand on the other answers with an example: Wikipedia has this neat diagram about cosets.
It portrays
- the group $G = mathbb Z_8$ under addition,
- its subgroup $H = {0, 4}$,
- all cosets of $H$ in $G$.
Clearly, only $H$ contains the identity element $0$, so only $H$ is a group. The cosets $1+H$ and $2+H$ and $3+H$ don’t contain $0$, so they fail to satisfy this important group property.
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
$endgroup$
add a comment |
$begingroup$
To expand on the other answers with an example: Wikipedia has this neat diagram about cosets.
It portrays
- the group $G = mathbb Z_8$ under addition,
- its subgroup $H = {0, 4}$,
- all cosets of $H$ in $G$.
Clearly, only $H$ contains the identity element $0$, so only $H$ is a group. The cosets $1+H$ and $2+H$ and $3+H$ don’t contain $0$, so they fail to satisfy this important group property.
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
$endgroup$
To expand on the other answers with an example: Wikipedia has this neat diagram about cosets.
It portrays
- the group $G = mathbb Z_8$ under addition,
- its subgroup $H = {0, 4}$,
- all cosets of $H$ in $G$.
Clearly, only $H$ contains the identity element $0$, so only $H$ is a group. The cosets $1+H$ and $2+H$ and $3+H$ don’t contain $0$, so they fail to satisfy this important group property.
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
answered Mar 18 '16 at 10:27
LynnLynn
2,5851526
2,5851526
add a comment |
add a comment |
$begingroup$
No, Let $(G,circ ,e)$ be our group, Note that for every $gin G$ and $Hle G$ s.t $gneq e$, We get that $enotin gH$ And therefor cannot be a subgroup of $H$ nor $G$ (the same goes for left cosets)
answered Mar 18 '16 at 10:16
shasha
987615
$endgroup$
add a comment |
$begingroup$
No, Let $(G,circ ,e)$ be our group, Note that for every $gin G$ and $Hle G$ s.t $gneq e$, We get that $enotin gH$ And therefor cannot be a subgroup of $H$ nor $G$ (the same goes for left cosets)
answered Mar 18 '16 at 10:16
shasha
987615
$endgroup$
add a comment |
$begingroup$
No, Let $(G,circ ,e)$ be our group, Note that for every $gin G$ and $Hle G$ s.t $gneq e$, We get that $enotin gH$ And therefor cannot be a subgroup of $H$ nor $G$ (the same goes for left cosets)
answered Mar 18 '16 at 10:16
shasha
987615
$endgroup$
No, Let $(G,circ ,e)$ be our group, Note that for every $gin G$ and $Hle G$ s.t $gneq e$, We get that $enotin gH$ And therefor cannot be a subgroup of $H$ nor $G$ (the same goes for left cosets)
answered Mar 18 '16 at 10:16
shasha
987615
answered Mar 18 '16 at 10:16
shasha
987615
answered Mar 18 '16 at 10:16
shasha
987615
answered Mar 18 '16 at 10:16
shasha
987615
987615
add a comment |
add a comment |
$begingroup$
If $H leq G$ then $aHbH=abH quad forall a,bin G$ if and only if $H$ is a normal subgroup of $G$ i.e $aHa^{-1}=H quad forall ain G$ .
For a counter example $ langle (1,2)rangle leq S_3 $.
$endgroup$
$begingroup$
I thought that he,she had asked that when cosets make a group :)
$endgroup$
– user217174
Mar 18 '16 at 14:20
add a comment |
$begingroup$
If $H leq G$ then $aHbH=abH quad forall a,bin G$ if and only if $H$ is a normal subgroup of $G$ i.e $aHa^{-1}=H quad forall ain G$ .
For a counter example $ langle (1,2)rangle leq S_3 $.
$endgroup$
$begingroup$
I thought that he,she had asked that when cosets make a group :)
$endgroup$
– user217174
Mar 18 '16 at 14:20
add a comment |
$begingroup$
If $H leq G$ then $aHbH=abH quad forall a,bin G$ if and only if $H$ is a normal subgroup of $G$ i.e $aHa^{-1}=H quad forall ain G$ .
For a counter example $ langle (1,2)rangle leq S_3 $.
$endgroup$
If $H leq G$ then $aHbH=abH quad forall a,bin G$ if and only if $H$ is a normal subgroup of $G$ i.e $aHa^{-1}=H quad forall ain G$ .
For a counter example $ langle (1,2)rangle leq S_3 $.
edited Mar 18 '16 at 10:20
answered Mar 18 '16 at 10:14
user217174
$begingroup$
I thought that he,she had asked that when cosets make a group :)
$endgroup$
– user217174
Mar 18 '16 at 14:20
add a comment |
$begingroup$
I thought that he,she had asked that when cosets make a group :)
$endgroup$
– user217174
Mar 18 '16 at 14:20
$begingroup$
I thought that he,she had asked that when cosets make a group :)
$endgroup$
– user217174
Mar 18 '16 at 14:20
$begingroup$
I thought that he,she had asked that when cosets make a group :)
$endgroup$
– user217174
Mar 18 '16 at 14:20
add a comment |
5
$begingroup$
No. In fact, only one of the cosets contains the identity element.
$endgroup$
– Darío G
Mar 18 '16 at 10:11