How to construct a subgroup.
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Consider the 2×2 matrices that (with respect to the standard basis of R2) represent rotation around the origin over 120 degrees and reflection in the x-axis. Construct the smallest possible subgroup of GL2(R) containing these two matrices. Is the resulting group abelian? Find the order of each element of this group.
How do I have to start? Never got this kind of questions before, the lecturer gave me a terrible explanation and there's nothing to find in the lecture notes.
group-theory abelian-groups
asked Nov 15 at 16:33
Peter van de Berg
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Consider the 2×2 matrices that (with respect to the standard basis of R2) represent rotation around the origin over 120 degrees and reflection in the x-axis. Construct the smallest possible subgroup of GL2(R) containing these two matrices. Is the resulting group abelian? Find the order of each element of this group.
How do I have to start? Never got this kind of questions before, the lecturer gave me a terrible explanation and there's nothing to find in the lecture notes.
group-theory abelian-groups
asked Nov 15 at 16:33
Peter van de Berg
208
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider the 2×2 matrices that (with respect to the standard basis of R2) represent rotation around the origin over 120 degrees and reflection in the x-axis. Construct the smallest possible subgroup of GL2(R) containing these two matrices. Is the resulting group abelian? Find the order of each element of this group.
How do I have to start? Never got this kind of questions before, the lecturer gave me a terrible explanation and there's nothing to find in the lecture notes.
group-theory abelian-groups
asked Nov 15 at 16:33
Peter van de Berg
208
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Consider the 2×2 matrices that (with respect to the standard basis of R2) represent rotation around the origin over 120 degrees and reflection in the x-axis. Construct the smallest possible subgroup of GL2(R) containing these two matrices. Is the resulting group abelian? Find the order of each element of this group.
How do I have to start? Never got this kind of questions before, the lecturer gave me a terrible explanation and there's nothing to find in the lecture notes.
group-theory abelian-groups
group-theory abelian-groups
asked Nov 15 at 16:33
Peter van de Berg
208
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 16:33
Peter van de Berg
208
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 16:33
Peter van de Berg
208
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 16:33
Peter van de Berg
208
asked Nov 15 at 16:33
Peter van de Berg
208
208
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
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0
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You start exactly as the question describes; first write down these matrices explicitly. Then look at the subgroup they generate; what do the products of these matrices look like? And their inverses?
While fiddling around, I would advise you to keep in mind the geometric picture of how the matrices act on $Bbb{R}^2$. That avoids a lot of computations.
EDIT: To make the fiddling around part more concrete; consider an equilateral triangle in $Bbb{R}^2$ with one vertex on the $x$-axis, and the other two symmetrically above and below the $x$-axis. Now it is easy to verify (both visually and by linear algebra) that the given matrices permute the vertices of the triangle. How many ways are there to permute the vertices in total? And can you get all permutations by applying some combination of these matrices?
answered Nov 15 at 16:42
Servaes
20.6k33789
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You start exactly as the question describes; first write down these matrices explicitly. Then look at the subgroup they generate; what do the products of these matrices look like? And their inverses?
While fiddling around, I would advise you to keep in mind the geometric picture of how the matrices act on $Bbb{R}^2$. That avoids a lot of computations.
EDIT: To make the fiddling around part more concrete; consider an equilateral triangle in $Bbb{R}^2$ with one vertex on the $x$-axis, and the other two symmetrically above and below the $x$-axis. Now it is easy to verify (both visually and by linear algebra) that the given matrices permute the vertices of the triangle. How many ways are there to permute the vertices in total? And can you get all permutations by applying some combination of these matrices?
answered Nov 15 at 16:42
Servaes
20.6k33789
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
add a comment |
up vote
0
down vote
You start exactly as the question describes; first write down these matrices explicitly. Then look at the subgroup they generate; what do the products of these matrices look like? And their inverses?
While fiddling around, I would advise you to keep in mind the geometric picture of how the matrices act on $Bbb{R}^2$. That avoids a lot of computations.
EDIT: To make the fiddling around part more concrete; consider an equilateral triangle in $Bbb{R}^2$ with one vertex on the $x$-axis, and the other two symmetrically above and below the $x$-axis. Now it is easy to verify (both visually and by linear algebra) that the given matrices permute the vertices of the triangle. How many ways are there to permute the vertices in total? And can you get all permutations by applying some combination of these matrices?
answered Nov 15 at 16:42
Servaes
20.6k33789
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
add a comment |
up vote
0
down vote
up vote
0
down vote
You start exactly as the question describes; first write down these matrices explicitly. Then look at the subgroup they generate; what do the products of these matrices look like? And their inverses?
While fiddling around, I would advise you to keep in mind the geometric picture of how the matrices act on $Bbb{R}^2$. That avoids a lot of computations.
EDIT: To make the fiddling around part more concrete; consider an equilateral triangle in $Bbb{R}^2$ with one vertex on the $x$-axis, and the other two symmetrically above and below the $x$-axis. Now it is easy to verify (both visually and by linear algebra) that the given matrices permute the vertices of the triangle. How many ways are there to permute the vertices in total? And can you get all permutations by applying some combination of these matrices?
answered Nov 15 at 16:42
Servaes
20.6k33789
You start exactly as the question describes; first write down these matrices explicitly. Then look at the subgroup they generate; what do the products of these matrices look like? And their inverses?
While fiddling around, I would advise you to keep in mind the geometric picture of how the matrices act on $Bbb{R}^2$. That avoids a lot of computations.
EDIT: To make the fiddling around part more concrete; consider an equilateral triangle in $Bbb{R}^2$ with one vertex on the $x$-axis, and the other two symmetrically above and below the $x$-axis. Now it is easy to verify (both visually and by linear algebra) that the given matrices permute the vertices of the triangle. How many ways are there to permute the vertices in total? And can you get all permutations by applying some combination of these matrices?
answered Nov 15 at 16:42
Servaes
20.6k33789
edited Nov 15 at 17:13
answered Nov 15 at 16:42
Servaes
20.6k33789
answered Nov 15 at 16:42
Servaes
20.6k33789
answered Nov 15 at 16:42
Servaes
20.6k33789
20.6k33789
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
add a comment |
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
I shall try it. Thanks. Only not sure if I am capable of doing this because the only courses I had before were basic calculus courses. (first-year mathematician). Never seen something like this before and I am stuck on the question the whole day already.
– Peter van de Berg
Nov 15 at 16:46
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Then I hope you have at least seen the definition of a group, and what 'abelian' and 'order' mean, as these don't usually show up in basic calculus courses.
– Servaes
Nov 15 at 16:48
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Yes, I found those defenitions on the internet. Problem is not knowing how to start, I tried youtube for visualization the problem, but that did not help me either.
– Peter van de Berg
Nov 15 at 16:51
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Assuming from your name that you can read Dutch, I can recommend a few sections from this reader; for the definitions see the first two pages of chapter 2. For an illustrative example, see section 2 of chapter 1 (symmetrieën van de ruit). This section treats precisely your question (except that you might want to consider an equilateral triangle in stead of a diamond) and assumes no prior knowledge of groups.
– Servaes
Nov 15 at 17:03
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
Yes I can read Dutch. Thankyou, I will take a look at it.
– Peter van de Berg
Nov 15 at 17:09
add a comment |
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
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