Finite Presentation of a subgroup
$begingroup$
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
12.8k22243
asked Jan 4 at 22:36
Jack CopperJack Copper
211
$endgroup$
add a comment |
$begingroup$
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
12.8k22243
asked Jan 4 at 22:36
Jack CopperJack Copper
211
$endgroup$
1
$begingroup$
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
$endgroup$
– Derek Holt
Jan 5 at 14:16
$begingroup$
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.
$endgroup$
– Jack Copper
Jan 5 at 19:47
$begingroup$
Ah, that's a different method, and I will leave someone else to help you with that.
$endgroup$
– Derek Holt
Jan 5 at 20:18
$begingroup$
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...
$endgroup$
– Moishe Kohan
Jan 5 at 23:03
add a comment |
$begingroup$
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
12.8k22243
asked Jan 4 at 22:36
Jack CopperJack Copper
211
$endgroup$
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
12.8k22243
asked Jan 4 at 22:36
Jack CopperJack Copper
211
edited Jan 5 at 20:26
André 3000
12.8k22243
asked Jan 4 at 22:36
Jack CopperJack Copper
211
edited Jan 5 at 20:26
André 3000
12.8k22243
edited Jan 5 at 20:26
André 3000
12.8k22243
edited Jan 5 at 20:26
André 3000
12.8k22243
12.8k22243
asked Jan 4 at 22:36
Jack CopperJack Copper
211
asked Jan 4 at 22:36
Jack CopperJack Copper
211
asked Jan 4 at 22:36
Jack CopperJack Copper
211
211
1
$begingroup$
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
$endgroup$
– Derek Holt
Jan 5 at 14:16
$begingroup$
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.
$endgroup$
– Jack Copper
Jan 5 at 19:47
$begingroup$
Ah, that's a different method, and I will leave someone else to help you with that.
$endgroup$
– Derek Holt
Jan 5 at 20:18
$begingroup$
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...
$endgroup$
– Moishe Kohan
Jan 5 at 23:03
add a comment |
1
$begingroup$
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
$endgroup$
– Derek Holt
Jan 5 at 14:16
$begingroup$
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.
$endgroup$
– Jack Copper
Jan 5 at 19:47
$begingroup$
Ah, that's a different method, and I will leave someone else to help you with that.
$endgroup$
– Derek Holt
Jan 5 at 20:18
$begingroup$
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...
$endgroup$
– Moishe Kohan
Jan 5 at 23:03
1
1
$begingroup$
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
$endgroup$
– Derek Holt
Jan 5 at 14:16
$begingroup$
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
$endgroup$
– Derek Holt
Jan 5 at 14:16
$begingroup$
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.
$endgroup$
– Jack Copper
Jan 5 at 19:47
$begingroup$
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.
$endgroup$
– Jack Copper
Jan 5 at 19:47
$begingroup$
Ah, that's a different method, and I will leave someone else to help you with that.
$endgroup$
– Derek Holt
Jan 5 at 20:18
$begingroup$
Ah, that's a different method, and I will leave someone else to help you with that.
$endgroup$
– Derek Holt
Jan 5 at 20:18
$begingroup$
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...
$endgroup$
– Moishe Kohan
Jan 5 at 23:03
$begingroup$
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...
$endgroup$
– Moishe Kohan
Jan 5 at 23:03
add a comment |
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$begingroup$
You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations.
$endgroup$
– Derek Holt
Jan 5 at 14:16
$begingroup$
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.
$endgroup$
– Jack Copper
Jan 5 at 19:47
$begingroup$
Ah, that's a different method, and I will leave someone else to help you with that.
$endgroup$
– Derek Holt
Jan 5 at 20:18
$begingroup$
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...
$endgroup$
– Moishe Kohan
Jan 5 at 23:03