Mixing
Find representatives for cosets. [closed]
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Let $G$ be a finite group and $H$ is a subgroup of $G$ where $[G:H]$ is $N$.
Prove that there exists a subset $A$ of $G$ containing $N$ elements such that, $xH= yH $ implies $x=y$ and $Hx= Hy$ implies $x=y$ for all $x,y$ in $A$.
This a problem from Basic Algebra by Jacobson.
I dont have an idea how to start. Kindly provide hints.
group-theory finite-groups
edited Jul 31 at 10:48
Alan Wang
4,088930
asked Jul 31 at 10:04
tony
1289
closed as off-topic by Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne Aug 1 at 0:47
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." – Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne
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up vote
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favorite
Let $G$ be a finite group and $H$ is a subgroup of $G$ where $[G:H]$ is $N$.
Prove that there exists a subset $A$ of $G$ containing $N$ elements such that, $xH= yH $ implies $x=y$ and $Hx= Hy$ implies $x=y$ for all $x,y$ in $A$.
This a problem from Basic Algebra by Jacobson.
I dont have an idea how to start. Kindly provide hints.
group-theory finite-groups
edited Jul 31 at 10:48
Alan Wang
4,088930
asked Jul 31 at 10:04
tony
1289
closed as off-topic by Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne Aug 1 at 0:47
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." – Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne
4
It is a consequence of Hall's Marriage Theorem. Make a bipartite graph with vertices consisting of left and right cosets, where a left coset is joined to a right coset if and only if they have nontrivial intersection. Then apply the marriage theorem to get a matching.
– Derek Holt
Jul 31 at 10:46
@DerekHolt I've totally misread the question. I think you should post that as an answer.
– freakish
Jul 31 at 11:42
@freakish Maybe later! I am reluctant to provide detailed answers to questions where the poster has made no effort. So for the moment I have just provided hints, as requested. In any case similar questions have been asked before.
– Derek Holt
Jul 31 at 12:38
Do you know of any solution without Hall theorem?
– tony
Jul 31 at 13:19
No. I have searched around and all of the references mention it as an application of the marriage theorem. Each right coset gets to marry a left coset with which it has something in common - that all sounds very suitable!
– Derek Holt
Jul 31 at 15:51
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $G$ be a finite group and $H$ is a subgroup of $G$ where $[G:H]$ is $N$.
Prove that there exists a subset $A$ of $G$ containing $N$ elements such that, $xH= yH $ implies $x=y$ and $Hx= Hy$ implies $x=y$ for all $x,y$ in $A$.
This a problem from Basic Algebra by Jacobson.
I dont have an idea how to start. Kindly provide hints.
group-theory finite-groups
edited Jul 31 at 10:48
Alan Wang
4,088930
asked Jul 31 at 10:04
tony
1289
Let $G$ be a finite group and $H$ is a subgroup of $G$ where $[G:H]$ is $N$.
Prove that there exists a subset $A$ of $G$ containing $N$ elements such that, $xH= yH $ implies $x=y$ and $Hx= Hy$ implies $x=y$ for all $x,y$ in $A$.
This a problem from Basic Algebra by Jacobson.
I dont have an idea how to start. Kindly provide hints.
group-theory finite-groups
edited Jul 31 at 10:48
Alan Wang
4,088930
asked Jul 31 at 10:04
tony
1289
edited Jul 31 at 10:48
Alan Wang
4,088930
edited Jul 31 at 10:48
Alan Wang
4,088930
edited Jul 31 at 10:48
Alan Wang
4,088930
4,088930
asked Jul 31 at 10:04
tony
1289
asked Jul 31 at 10:04
tony
1289
asked Jul 31 at 10:04
tony
1289
1289
closed as off-topic by Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne Aug 1 at 0:47
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." – Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne
closed as off-topic by Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne Aug 1 at 0:47
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." – Derek Holt, John Ma, Mostafa Ayaz, amWhy, Isaac Browne
4
It is a consequence of Hall's Marriage Theorem. Make a bipartite graph with vertices consisting of left and right cosets, where a left coset is joined to a right coset if and only if they have nontrivial intersection. Then apply the marriage theorem to get a matching.
– Derek Holt
Jul 31 at 10:46
@DerekHolt I've totally misread the question. I think you should post that as an answer.
– freakish
Jul 31 at 11:42
@freakish Maybe later! I am reluctant to provide detailed answers to questions where the poster has made no effort. So for the moment I have just provided hints, as requested. In any case similar questions have been asked before.
– Derek Holt
Jul 31 at 12:38
Do you know of any solution without Hall theorem?
– tony
Jul 31 at 13:19
No. I have searched around and all of the references mention it as an application of the marriage theorem. Each right coset gets to marry a left coset with which it has something in common - that all sounds very suitable!
– Derek Holt
Jul 31 at 15:51
add a comment |Â
4
It is a consequence of Hall's Marriage Theorem. Make a bipartite graph with vertices consisting of left and right cosets, where a left coset is joined to a right coset if and only if they have nontrivial intersection. Then apply the marriage theorem to get a matching.
– Derek Holt
Jul 31 at 10:46
@DerekHolt I've totally misread the question. I think you should post that as an answer.
– freakish
Jul 31 at 11:42
@freakish Maybe later! I am reluctant to provide detailed answers to questions where the poster has made no effort. So for the moment I have just provided hints, as requested. In any case similar questions have been asked before.
– Derek Holt
Jul 31 at 12:38
Do you know of any solution without Hall theorem?
– tony
Jul 31 at 13:19
No. I have searched around and all of the references mention it as an application of the marriage theorem. Each right coset gets to marry a left coset with which it has something in common - that all sounds very suitable!
– Derek Holt
Jul 31 at 15:51
4
4
It is a consequence of Hall's Marriage Theorem. Make a bipartite graph with vertices consisting of left and right cosets, where a left coset is joined to a right coset if and only if they have nontrivial intersection. Then apply the marriage theorem to get a matching.
– Derek Holt
Jul 31 at 10:46
It is a consequence of Hall's Marriage Theorem. Make a bipartite graph with vertices consisting of left and right cosets, where a left coset is joined to a right coset if and only if they have nontrivial intersection. Then apply the marriage theorem to get a matching.
– Derek Holt
Jul 31 at 10:46
@DerekHolt I've totally misread the question. I think you should post that as an answer.
– freakish
Jul 31 at 11:42
@DerekHolt I've totally misread the question. I think you should post that as an answer.
– freakish
Jul 31 at 11:42
@freakish Maybe later! I am reluctant to provide detailed answers to questions where the poster has made no effort. So for the moment I have just provided hints, as requested. In any case similar questions have been asked before.
– Derek Holt
Jul 31 at 12:38
@freakish Maybe later! I am reluctant to provide detailed answers to questions where the poster has made no effort. So for the moment I have just provided hints, as requested. In any case similar questions have been asked before.
– Derek Holt
Jul 31 at 12:38
Do you know of any solution without Hall theorem?
– tony
Jul 31 at 13:19
Do you know of any solution without Hall theorem?
– tony
Jul 31 at 13:19
No. I have searched around and all of the references mention it as an application of the marriage theorem. Each right coset gets to marry a left coset with which it has something in common - that all sounds very suitable!
– Derek Holt
Jul 31 at 15:51
No. I have searched around and all of the references mention it as an application of the marriage theorem. Each right coset gets to marry a left coset with which it has something in common - that all sounds very suitable!
– Derek Holt
Jul 31 at 15:51
add a comment |Â
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4
It is a consequence of Hall's Marriage Theorem. Make a bipartite graph with vertices consisting of left and right cosets, where a left coset is joined to a right coset if and only if they have nontrivial intersection. Then apply the marriage theorem to get a matching.
– Derek Holt
Jul 31 at 10:46
@DerekHolt I've totally misread the question. I think you should post that as an answer.
– freakish
Jul 31 at 11:42
@freakish Maybe later! I am reluctant to provide detailed answers to questions where the poster has made no effort. So for the moment I have just provided hints, as requested. In any case similar questions have been asked before.
– Derek Holt
Jul 31 at 12:38
Do you know of any solution without Hall theorem?
– tony
Jul 31 at 13:19
No. I have searched around and all of the references mention it as an application of the marriage theorem. Each right coset gets to marry a left coset with which it has something in common - that all sounds very suitable!
– Derek Holt
Jul 31 at 15:51