Proving a group $G$ with presentation $langle a,bmid abrangle$ is isomorphic to $Bbb Z$.
Given that $ab=e$, we know $b =a^{-1}$. Since $G = langle a,brangle$ this implies $langle a,brangle=langle a,a^{-1}rangle=langle arangle=G$ (as elements that generate the group). The presentation rewritten in terms of $a$ is trivial, i.e. $langle a,a^{-1}mid aa^{-1}=erangle$, so $G$ is a free group on the generator $a$, which is isomorphic to $mathbb{Z}$.
I have just learned about group presentations and I am also wondering if there is a general method for identifying the isomorphism classes of a group given its presentation.
group-theory proof-verification group-presentation combinatorial-group-theory
edited Nov 30 '18 at 2:52
Shaun
8,820113681
asked May 5 '16 at 19:19
user52969user52969
580210
add a comment |
Given that $ab=e$, we know $b =a^{-1}$. Since $G = langle a,brangle$ this implies $langle a,brangle=langle a,a^{-1}rangle=langle arangle=G$ (as elements that generate the group). The presentation rewritten in terms of $a$ is trivial, i.e. $langle a,a^{-1}mid aa^{-1}=erangle$, so $G$ is a free group on the generator $a$, which is isomorphic to $mathbb{Z}$.
I have just learned about group presentations and I am also wondering if there is a general method for identifying the isomorphism classes of a group given its presentation.
group-theory proof-verification group-presentation combinatorial-group-theory
edited Nov 30 '18 at 2:52
Shaun
8,820113681
asked May 5 '16 at 19:19
user52969user52969
580210
No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable.
– Derek Holt
May 5 '16 at 19:28
Does this mean the conclusion of the proof is wrong
– user52969
May 5 '16 at 19:41
4
No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations.
– Derek Holt
May 5 '16 at 19:46
add a comment |
Given that $ab=e$, we know $b =a^{-1}$. Since $G = langle a,brangle$ this implies $langle a,brangle=langle a,a^{-1}rangle=langle arangle=G$ (as elements that generate the group). The presentation rewritten in terms of $a$ is trivial, i.e. $langle a,a^{-1}mid aa^{-1}=erangle$, so $G$ is a free group on the generator $a$, which is isomorphic to $mathbb{Z}$.
I have just learned about group presentations and I am also wondering if there is a general method for identifying the isomorphism classes of a group given its presentation.
group-theory proof-verification group-presentation combinatorial-group-theory
edited Nov 30 '18 at 2:52
Shaun
8,820113681
asked May 5 '16 at 19:19
user52969user52969
580210
Given that $ab=e$, we know $b =a^{-1}$. Since $G = langle a,brangle$ this implies $langle a,brangle=langle a,a^{-1}rangle=langle arangle=G$ (as elements that generate the group). The presentation rewritten in terms of $a$ is trivial, i.e. $langle a,a^{-1}mid aa^{-1}=erangle$, so $G$ is a free group on the generator $a$, which is isomorphic to $mathbb{Z}$.
I have just learned about group presentations and I am also wondering if there is a general method for identifying the isomorphism classes of a group given its presentation.
group-theory proof-verification group-presentation combinatorial-group-theory
group-theory proof-verification group-presentation combinatorial-group-theory
edited Nov 30 '18 at 2:52
Shaun
8,820113681
asked May 5 '16 at 19:19
user52969user52969
580210
edited Nov 30 '18 at 2:52
Shaun
8,820113681
asked May 5 '16 at 19:19
user52969user52969
580210
edited Nov 30 '18 at 2:52
Shaun
8,820113681
edited Nov 30 '18 at 2:52
Shaun
8,820113681
edited Nov 30 '18 at 2:52
Shaun
8,820113681
8,820113681
asked May 5 '16 at 19:19
user52969user52969
580210
asked May 5 '16 at 19:19
user52969user52969
580210
asked May 5 '16 at 19:19
user52969user52969
580210
580210
No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable.
– Derek Holt
May 5 '16 at 19:28
Does this mean the conclusion of the proof is wrong
– user52969
May 5 '16 at 19:41
4
No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations.
– Derek Holt
May 5 '16 at 19:46
add a comment |
No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable.
– Derek Holt
May 5 '16 at 19:28
Does this mean the conclusion of the proof is wrong
– user52969
May 5 '16 at 19:41
4
No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations.
– Derek Holt
May 5 '16 at 19:46
No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable.
– Derek Holt
May 5 '16 at 19:28
No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable.
– Derek Holt
May 5 '16 at 19:28
Does this mean the conclusion of the proof is wrong
– user52969
May 5 '16 at 19:41
Does this mean the conclusion of the proof is wrong
– user52969
May 5 '16 at 19:41
4
4
No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations.
– Derek Holt
May 5 '16 at 19:46
No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations.
– Derek Holt
May 5 '16 at 19:46
add a comment |
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No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable.
– Derek Holt
May 5 '16 at 19:28
Does this mean the conclusion of the proof is wrong
– user52969
May 5 '16 at 19:41
4
No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations.
– Derek Holt
May 5 '16 at 19:46