Finite Presentation of a subgroup
I have the group $langle a,b mid a^3b^3rangle$ Now I send both $a$ and $b$ to the generator of $mathbb{Z}/3mathbb{Z}$. This gives a well-defined homomorphism from our group to $mathbb{Z}/3mathbb{Z}$ and I am asked to find a finite presentation of the kernel of this homomorphism. How do I generally tackle these kind of questions?
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
abstract-algebra group-theory geometric-group-theory group-presentation
edited Jan 5 at 20:26
André 3000
12.5k22042
asked Jan 4 at 22:36
Jack CopperJack Copper
161
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Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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I have the group $langle a,b mid a^3b^3rangle$ Now I send both $a$ and $b$ to the generator of $mathbb{Z}/3mathbb{Z}$. This gives a well-defined homomorphism from our group to $mathbb{Z}/3mathbb{Z}$ and I am asked to find a finite presentation of the kernel of this homomorphism. How do I generally tackle these kind of questions?
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
abstract-algebra group-theory geometric-group-theory group-presentation
edited Jan 5 at 20:26
André 3000
12.5k22042
asked Jan 4 at 22:36
Jack CopperJack Copper
161
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
– Derek Holt
Jan 5 at 14:16
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
– Jack Copper
Jan 5 at 19:47
Ah, that's a different method, and I will leave someone else to help you with that.
– Derek Holt
Jan 5 at 20:18
Hint: Start with the wedge of two circles $X=S^1 vee S^1$ and consider a homomorphism $h: pi_1(X)to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells...
– Moishe Cohen
Jan 5 at 23:03
add a comment |
I have the group $langle a,b mid a^3b^3rangle$ Now I send both $a$ and $b$ to the generator of $mathbb{Z}/3mathbb{Z}$. This gives a well-defined homomorphism from our group to $mathbb{Z}/3mathbb{Z}$ and I am asked to find a finite presentation of the kernel of this homomorphism. How do I generally tackle these kind of questions?
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
abstract-algebra group-theory geometric-group-theory group-presentation
edited Jan 5 at 20:26
André 3000
12.5k22042
asked Jan 4 at 22:36
Jack CopperJack Copper
161
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I have the group $langle a,b mid a^3b^3rangle$ Now I send both $a$ and $b$ to the generator of $mathbb{Z}/3mathbb{Z}$. This gives a well-defined homomorphism from our group to $mathbb{Z}/3mathbb{Z}$ and I am asked to find a finite presentation of the kernel of this homomorphism. How do I generally tackle these kind of questions?
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
abstract-algebra group-theory geometric-group-theory group-presentation
abstract-algebra group-theory geometric-group-theory group-presentation
edited Jan 5 at 20:26
André 3000
12.5k22042
asked Jan 4 at 22:36
Jack CopperJack Copper
161
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 5 at 20:26
André 3000
12.5k22042
asked Jan 4 at 22:36
Jack CopperJack Copper
161
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 5 at 20:26
André 3000
12.5k22042
edited Jan 5 at 20:26
André 3000
12.5k22042
edited Jan 5 at 20:26
André 3000
12.5k22042
12.5k22042
asked Jan 4 at 22:36
Jack CopperJack Copper
161
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 22:36
Jack CopperJack Copper
161
asked Jan 4 at 22:36
Jack CopperJack Copper
161
161
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Jack Copper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
– Derek Holt
Jan 5 at 14:16
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
– Jack Copper
Jan 5 at 19:47
Ah, that's a different method, and I will leave someone else to help you with that.
– Derek Holt
Jan 5 at 20:18
Hint: Start with the wedge of two circles $X=S^1 vee S^1$ and consider a homomorphism $h: pi_1(X)to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells...
– Moishe Cohen
Jan 5 at 23:03
add a comment |
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
– Derek Holt
Jan 5 at 14:16
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
– Jack Copper
Jan 5 at 19:47
Ah, that's a different method, and I will leave someone else to help you with that.
– Derek Holt
Jan 5 at 20:18
Hint: Start with the wedge of two circles $X=S^1 vee S^1$ and consider a homomorphism $h: pi_1(X)to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells...
– Moishe Cohen
Jan 5 at 23:03
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
– Derek Holt
Jan 5 at 14:16
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
– Derek Holt
Jan 5 at 14:16
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
– Jack Copper
Jan 5 at 19:47
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
– Jack Copper
Jan 5 at 19:47
Ah, that's a different method, and I will leave someone else to help you with that.
– Derek Holt
Jan 5 at 20:18
Ah, that's a different method, and I will leave someone else to help you with that.
– Derek Holt
Jan 5 at 20:18
Hint: Start with the wedge of two circles $X=S^1 vee S^1$ and consider a homomorphism $h: pi_1(X)to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells...
– Moishe Cohen
Jan 5 at 23:03
Hint: Start with the wedge of two circles $X=S^1 vee S^1$ and consider a homomorphism $h: pi_1(X)to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells...
– Moishe Cohen
Jan 5 at 23:03
add a comment |
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You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
– Derek Holt
Jan 5 at 14:16
I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.
– Jack Copper
Jan 5 at 19:47
Ah, that's a different method, and I will leave someone else to help you with that.
– Derek Holt
Jan 5 at 20:18
Hint: Start with the wedge of two circles $X=S^1 vee S^1$ and consider a homomorphism $h: pi_1(X)to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells...
– Moishe Cohen
Jan 5 at 23:03