Showing the Heisenberg group admits the presentation $langle A, B, Cmid AC=CA, BC=CB,...
Consider the subset of $SL_3(Bbb Z)$ consisting of matrices of the form
$$begin{pmatrix}
1 & a & c\
0 & 1 & b \
0 & 0 & 1
end{pmatrix}$$
for $a,b,cinBbb Z$.
I need to show that it admits the presentation:
$$langle A, B, Cmid AC=CA, BC=CB, ABA^{-1}B^{-1}=Crangle.$$
I am not sure what exactly I need to show. I tried to follow the definition given in class, but didn't really understand it.
Definition 1.36: Let $G$ be a group and $S$ a generating set. Let $Rsubseteq F(S)$. Denote by $pi$ the morphism $F(S)to G$. We say that $G$ admits the presentation $langle Smid Rrangle$ if $R$ normally generates $ker pi$, that is, if $kerpi$ is the smallest normal subgroup containing $R$.
Remark 1.38: We can also build a group with any presentation we choose: given a set $S$ and a set $R$ of words in $S$, the quotient $F(S)/langlelangle Rranglerangle$ obviously admits the presentation $langle Smid Rrangle$. Note that this is a way to specify a group algebraically, not geometrically.
group-theory geometric-group-theory group-presentation
edited Nov 29 '18 at 22:16
Shaun
8,820113681
asked Dec 5 '16 at 13:28
matanmatan
23319
|
show 5 more comments
Consider the subset of $SL_3(Bbb Z)$ consisting of matrices of the form
$$begin{pmatrix}
1 & a & c\
0 & 1 & b \
0 & 0 & 1
end{pmatrix}$$
for $a,b,cinBbb Z$.
I need to show that it admits the presentation:
$$langle A, B, Cmid AC=CA, BC=CB, ABA^{-1}B^{-1}=Crangle.$$
I am not sure what exactly I need to show. I tried to follow the definition given in class, but didn't really understand it.
Definition 1.36: Let $G$ be a group and $S$ a generating set. Let $Rsubseteq F(S)$. Denote by $pi$ the morphism $F(S)to G$. We say that $G$ admits the presentation $langle Smid Rrangle$ if $R$ normally generates $ker pi$, that is, if $kerpi$ is the smallest normal subgroup containing $R$.
Remark 1.38: We can also build a group with any presentation we choose: given a set $S$ and a set $R$ of words in $S$, the quotient $F(S)/langlelangle Rranglerangle$ obviously admits the presentation $langle Smid Rrangle$. Note that this is a way to specify a group algebraically, not geometrically.
group-theory geometric-group-theory group-presentation
edited Nov 29 '18 at 22:16
Shaun
8,820113681
asked Dec 5 '16 at 13:28
matanmatan
23319
3
Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten?
– Lee Mosher
Dec 5 '16 at 13:46
1
That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions.
– Lee Mosher
Dec 5 '16 at 14:35
4
The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k in {mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups.
– Derek Holt
Dec 5 '16 at 16:37
1
OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi)
– matan
Dec 5 '16 at 19:02
1
Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal.
– Moishe Cohen
Dec 5 '16 at 20:31
|
show 5 more comments
Consider the subset of $SL_3(Bbb Z)$ consisting of matrices of the form
$$begin{pmatrix}
1 & a & c\
0 & 1 & b \
0 & 0 & 1
end{pmatrix}$$
for $a,b,cinBbb Z$.
I need to show that it admits the presentation:
$$langle A, B, Cmid AC=CA, BC=CB, ABA^{-1}B^{-1}=Crangle.$$
I am not sure what exactly I need to show. I tried to follow the definition given in class, but didn't really understand it.
Definition 1.36: Let $G$ be a group and $S$ a generating set. Let $Rsubseteq F(S)$. Denote by $pi$ the morphism $F(S)to G$. We say that $G$ admits the presentation $langle Smid Rrangle$ if $R$ normally generates $ker pi$, that is, if $kerpi$ is the smallest normal subgroup containing $R$.
Remark 1.38: We can also build a group with any presentation we choose: given a set $S$ and a set $R$ of words in $S$, the quotient $F(S)/langlelangle Rranglerangle$ obviously admits the presentation $langle Smid Rrangle$. Note that this is a way to specify a group algebraically, not geometrically.
group-theory geometric-group-theory group-presentation
edited Nov 29 '18 at 22:16
Shaun
8,820113681
asked Dec 5 '16 at 13:28
matanmatan
23319
Consider the subset of $SL_3(Bbb Z)$ consisting of matrices of the form
$$begin{pmatrix}
1 & a & c\
0 & 1 & b \
0 & 0 & 1
end{pmatrix}$$
for $a,b,cinBbb Z$.
I need to show that it admits the presentation:
$$langle A, B, Cmid AC=CA, BC=CB, ABA^{-1}B^{-1}=Crangle.$$
I am not sure what exactly I need to show. I tried to follow the definition given in class, but didn't really understand it.
Definition 1.36: Let $G$ be a group and $S$ a generating set. Let $Rsubseteq F(S)$. Denote by $pi$ the morphism $F(S)to G$. We say that $G$ admits the presentation $langle Smid Rrangle$ if $R$ normally generates $ker pi$, that is, if $kerpi$ is the smallest normal subgroup containing $R$.
Remark 1.38: We can also build a group with any presentation we choose: given a set $S$ and a set $R$ of words in $S$, the quotient $F(S)/langlelangle Rranglerangle$ obviously admits the presentation $langle Smid Rrangle$. Note that this is a way to specify a group algebraically, not geometrically.
group-theory geometric-group-theory group-presentation
group-theory geometric-group-theory group-presentation
edited Nov 29 '18 at 22:16
Shaun
8,820113681
asked Dec 5 '16 at 13:28
matanmatan
23319
edited Nov 29 '18 at 22:16
Shaun
8,820113681
asked Dec 5 '16 at 13:28
matanmatan
23319
edited Nov 29 '18 at 22:16
Shaun
8,820113681
edited Nov 29 '18 at 22:16
Shaun
8,820113681
edited Nov 29 '18 at 22:16
Shaun
8,820113681
8,820113681
asked Dec 5 '16 at 13:28
matanmatan
23319
asked Dec 5 '16 at 13:28
matanmatan
23319
asked Dec 5 '16 at 13:28
matanmatan
23319
23319
3
Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten?
– Lee Mosher
Dec 5 '16 at 13:46
1
That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions.
– Lee Mosher
Dec 5 '16 at 14:35
4
The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k in {mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups.
– Derek Holt
Dec 5 '16 at 16:37
1
OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi)
– matan
Dec 5 '16 at 19:02
1
Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal.
– Moishe Cohen
Dec 5 '16 at 20:31
|
show 5 more comments
3
Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten?
– Lee Mosher
Dec 5 '16 at 13:46
1
That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions.
– Lee Mosher
Dec 5 '16 at 14:35
4
The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k in {mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups.
– Derek Holt
Dec 5 '16 at 16:37
1
OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi)
– matan
Dec 5 '16 at 19:02
1
Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal.
– Moishe Cohen
Dec 5 '16 at 20:31
3
3
Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten?
– Lee Mosher
Dec 5 '16 at 13:46
Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten?
– Lee Mosher
Dec 5 '16 at 13:46
1
1
That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions.
– Lee Mosher
Dec 5 '16 at 14:35
That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions.
– Lee Mosher
Dec 5 '16 at 14:35
4
4
The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k in {mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups.
– Derek Holt
Dec 5 '16 at 16:37
The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k in {mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups.
– Derek Holt
Dec 5 '16 at 16:37
1
1
OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi)
– matan
Dec 5 '16 at 19:02
OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi)
– matan
Dec 5 '16 at 19:02
1
1
Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal.
– Moishe Cohen
Dec 5 '16 at 20:31
Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal.
– Moishe Cohen
Dec 5 '16 at 20:31
|
show 5 more comments
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3
Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten?
– Lee Mosher
Dec 5 '16 at 13:46
1
That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions.
– Lee Mosher
Dec 5 '16 at 14:35
4
The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k in {mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups.
– Derek Holt
Dec 5 '16 at 16:37
1
OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi)
– matan
Dec 5 '16 at 19:02
1
Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal.
– Moishe Cohen
Dec 5 '16 at 20:31