Can you make the triviality of $langle a,b,c mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 rangle$ more...
$begingroup$
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,cin G$. If $aba^{-1} = b^2$, $bcb^{-1} = c^2$, and $cac^{-1} = a^2$, then $a = b = c = e$. Put another way, the group defined by generators and relations $langle a,b,c mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 rangle$ is the trivial group.
I came up with the following elementary, but ugly, proof:
The relations can be rewritten as (1) $ab = b^2a$, (2) $bc = c^2b$, (3) $ca = a^2c$.
Using (1), (2), and (3), we can rewrite $a^4bc = a^4c^2b = c^2ab = c^2b^2a$.
But we can also rewrite $a^4bc = b^{16}a^4c = b^{16}ca^2 = c^{2^{16}}b^{16}a^2$.
So $c^{2^{16}}b^{16}a^2 = c^2b^2a$. This implies $a = b^{-16}c^{2-2^{16}}b^2$.
Substituting for $a$ in (1) above, $b^{-16}c^{2(1-2^{15})}b^3 = b^{-14}c^{2(1-2^{15})}b^2$, and cancelling from both sides, $c^{2(1-2^{15})}b = b^{2}c^{2(1-2^{15})}$.
But now by (2), we have $bc^{1-2^{15}} = b^{2}c^{2(1-2^{15})}$, and $b = c^{2^{15}-1}$. But then $b$ and $c$ commute, so $bcb^{-1} = c^2$ implies $c = c^2$, and $c = e$. It then follows easily that $a = b = c = e$.
Question: Is there a better way to see this? i.e. a more abstract proof, or at least one that doesn't involve manipulating words of length $2^{16}$?
abstract-algebra group-theory group-presentation
edited Jan 4 at 6:14
Shaun
9,050113682
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
$endgroup$
|
show 3 more comments
$begingroup$
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,cin G$. If $aba^{-1} = b^2$, $bcb^{-1} = c^2$, and $cac^{-1} = a^2$, then $a = b = c = e$. Put another way, the group defined by generators and relations $langle a,b,c mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 rangle$ is the trivial group.
I came up with the following elementary, but ugly, proof:
The relations can be rewritten as (1) $ab = b^2a$, (2) $bc = c^2b$, (3) $ca = a^2c$.
Using (1), (2), and (3), we can rewrite $a^4bc = a^4c^2b = c^2ab = c^2b^2a$.
But we can also rewrite $a^4bc = b^{16}a^4c = b^{16}ca^2 = c^{2^{16}}b^{16}a^2$.
So $c^{2^{16}}b^{16}a^2 = c^2b^2a$. This implies $a = b^{-16}c^{2-2^{16}}b^2$.
Substituting for $a$ in (1) above, $b^{-16}c^{2(1-2^{15})}b^3 = b^{-14}c^{2(1-2^{15})}b^2$, and cancelling from both sides, $c^{2(1-2^{15})}b = b^{2}c^{2(1-2^{15})}$.
But now by (2), we have $bc^{1-2^{15}} = b^{2}c^{2(1-2^{15})}$, and $b = c^{2^{15}-1}$. But then $b$ and $c$ commute, so $bcb^{-1} = c^2$ implies $c = c^2$, and $c = e$. It then follows easily that $a = b = c = e$.
Question: Is there a better way to see this? i.e. a more abstract proof, or at least one that doesn't involve manipulating words of length $2^{16}$?
abstract-algebra group-theory group-presentation
edited Jan 4 at 6:14
Shaun
9,050113682
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
$endgroup$
5
$begingroup$
A slightly relevant remark, if there are four generators with similar relations, the group will be infinite. That is, $⟨a,b,c,d∣aba^{−1}=b^2, bcb^{−1}=c^2,cdc^{−1}=d^2, dad^{−1}=a^2⟩$ is not finite. c.f. Serre's Trees Page 9.
$endgroup$
– userabc
Jan 4 at 5:02
1
$begingroup$
@userabc Thanks for the comment - this is actually the very next remark in the paper I linked to. But the authors just call it a "well-known fact", so it's nice to have the reference to Serre's book. And I see that my question is Exercise 1) on p. 10 of Trees.
$endgroup$
– Alex Kruckman
Jan 4 at 5:14
1
$begingroup$
... and now that I can search for an exercise number in a well-known book, I find that there are a number of questions on this site about the same group. For example, Jim Belk gave almost an identical proof to the one I found here, and Martin Brandenburg asked essentially the same question I'm asking here. Together, this evidence makes me think that there's unlikely to be a nicer proof. It would be reasonable to close this question as a duplicate.
$endgroup$
– Alex Kruckman
Jan 4 at 5:26
$begingroup$
There is another proof here.
$endgroup$
– Derek Holt
Jan 4 at 8:58
3
$begingroup$
Once you have proved that $a in langle b,c rangle$, you can use the more conceptual argument given by Bhaskar Vashishth in the linked proof, that $G$ is perfect, but $G = langle b,c rangle$ is solvable, so $G$ is trivial.
$endgroup$
– Derek Holt
Jan 4 at 15:05
|
show 3 more comments
$begingroup$
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,cin G$. If $aba^{-1} = b^2$, $bcb^{-1} = c^2$, and $cac^{-1} = a^2$, then $a = b = c = e$. Put another way, the group defined by generators and relations $langle a,b,c mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 rangle$ is the trivial group.
I came up with the following elementary, but ugly, proof:
The relations can be rewritten as (1) $ab = b^2a$, (2) $bc = c^2b$, (3) $ca = a^2c$.
Using (1), (2), and (3), we can rewrite $a^4bc = a^4c^2b = c^2ab = c^2b^2a$.
But we can also rewrite $a^4bc = b^{16}a^4c = b^{16}ca^2 = c^{2^{16}}b^{16}a^2$.
So $c^{2^{16}}b^{16}a^2 = c^2b^2a$. This implies $a = b^{-16}c^{2-2^{16}}b^2$.
Substituting for $a$ in (1) above, $b^{-16}c^{2(1-2^{15})}b^3 = b^{-14}c^{2(1-2^{15})}b^2$, and cancelling from both sides, $c^{2(1-2^{15})}b = b^{2}c^{2(1-2^{15})}$.
But now by (2), we have $bc^{1-2^{15}} = b^{2}c^{2(1-2^{15})}$, and $b = c^{2^{15}-1}$. But then $b$ and $c$ commute, so $bcb^{-1} = c^2$ implies $c = c^2$, and $c = e$. It then follows easily that $a = b = c = e$.
Question: Is there a better way to see this? i.e. a more abstract proof, or at least one that doesn't involve manipulating words of length $2^{16}$?
abstract-algebra group-theory group-presentation
edited Jan 4 at 6:14
Shaun
9,050113682
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
$endgroup$
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,cin G$. If $aba^{-1} = b^2$, $bcb^{-1} = c^2$, and $cac^{-1} = a^2$, then $a = b = c = e$. Put another way, the group defined by generators and relations $langle a,b,c mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 rangle$ is the trivial group.
I came up with the following elementary, but ugly, proof:
The relations can be rewritten as (1) $ab = b^2a$, (2) $bc = c^2b$, (3) $ca = a^2c$.
Using (1), (2), and (3), we can rewrite $a^4bc = a^4c^2b = c^2ab = c^2b^2a$.
But we can also rewrite $a^4bc = b^{16}a^4c = b^{16}ca^2 = c^{2^{16}}b^{16}a^2$.
So $c^{2^{16}}b^{16}a^2 = c^2b^2a$. This implies $a = b^{-16}c^{2-2^{16}}b^2$.
Substituting for $a$ in (1) above, $b^{-16}c^{2(1-2^{15})}b^3 = b^{-14}c^{2(1-2^{15})}b^2$, and cancelling from both sides, $c^{2(1-2^{15})}b = b^{2}c^{2(1-2^{15})}$.
But now by (2), we have $bc^{1-2^{15}} = b^{2}c^{2(1-2^{15})}$, and $b = c^{2^{15}-1}$. But then $b$ and $c$ commute, so $bcb^{-1} = c^2$ implies $c = c^2$, and $c = e$. It then follows easily that $a = b = c = e$.
Question: Is there a better way to see this? i.e. a more abstract proof, or at least one that doesn't involve manipulating words of length $2^{16}$?
abstract-algebra group-theory group-presentation
abstract-algebra group-theory group-presentation
edited Jan 4 at 6:14
Shaun
9,050113682
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
edited Jan 4 at 6:14
Shaun
9,050113682
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
edited Jan 4 at 6:14
Shaun
9,050113682
edited Jan 4 at 6:14
Shaun
9,050113682
edited Jan 4 at 6:14
Shaun
9,050113682
9,050113682
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
asked Jan 4 at 4:13
Alex KruckmanAlex Kruckman
27.1k22556
27.1k22556
5
$begingroup$
A slightly relevant remark, if there are four generators with similar relations, the group will be infinite. That is, $⟨a,b,c,d∣aba^{−1}=b^2, bcb^{−1}=c^2,cdc^{−1}=d^2, dad^{−1}=a^2⟩$ is not finite. c.f. Serre's Trees Page 9.
$endgroup$
– userabc
Jan 4 at 5:02
1
$begingroup$
@userabc Thanks for the comment - this is actually the very next remark in the paper I linked to. But the authors just call it a "well-known fact", so it's nice to have the reference to Serre's book. And I see that my question is Exercise 1) on p. 10 of Trees.
$endgroup$
– Alex Kruckman
Jan 4 at 5:14
1
$begingroup$
... and now that I can search for an exercise number in a well-known book, I find that there are a number of questions on this site about the same group. For example, Jim Belk gave almost an identical proof to the one I found here, and Martin Brandenburg asked essentially the same question I'm asking here. Together, this evidence makes me think that there's unlikely to be a nicer proof. It would be reasonable to close this question as a duplicate.
$endgroup$
– Alex Kruckman
Jan 4 at 5:26
$begingroup$
There is another proof here.
$endgroup$
– Derek Holt
Jan 4 at 8:58
3
$begingroup$
Once you have proved that $a in langle b,c rangle$, you can use the more conceptual argument given by Bhaskar Vashishth in the linked proof, that $G$ is perfect, but $G = langle b,c rangle$ is solvable, so $G$ is trivial.
$endgroup$
– Derek Holt
Jan 4 at 15:05
|
show 3 more comments
5
$begingroup$
A slightly relevant remark, if there are four generators with similar relations, the group will be infinite. That is, $⟨a,b,c,d∣aba^{−1}=b^2, bcb^{−1}=c^2,cdc^{−1}=d^2, dad^{−1}=a^2⟩$ is not finite. c.f. Serre's Trees Page 9.
$endgroup$
– userabc
Jan 4 at 5:02
1
$begingroup$
@userabc Thanks for the comment - this is actually the very next remark in the paper I linked to. But the authors just call it a "well-known fact", so it's nice to have the reference to Serre's book. And I see that my question is Exercise 1) on p. 10 of Trees.
$endgroup$
– Alex Kruckman
Jan 4 at 5:14
1
$begingroup$
... and now that I can search for an exercise number in a well-known book, I find that there are a number of questions on this site about the same group. For example, Jim Belk gave almost an identical proof to the one I found here, and Martin Brandenburg asked essentially the same question I'm asking here. Together, this evidence makes me think that there's unlikely to be a nicer proof. It would be reasonable to close this question as a duplicate.
$endgroup$
– Alex Kruckman
Jan 4 at 5:26
$begingroup$
There is another proof here.
$endgroup$
– Derek Holt
Jan 4 at 8:58
3
$begingroup$
Once you have proved that $a in langle b,c rangle$, you can use the more conceptual argument given by Bhaskar Vashishth in the linked proof, that $G$ is perfect, but $G = langle b,c rangle$ is solvable, so $G$ is trivial.
$endgroup$
– Derek Holt
Jan 4 at 15:05
5
5
$begingroup$
A slightly relevant remark, if there are four generators with similar relations, the group will be infinite. That is, $⟨a,b,c,d∣aba^{−1}=b^2, bcb^{−1}=c^2,cdc^{−1}=d^2, dad^{−1}=a^2⟩$ is not finite. c.f. Serre's Trees Page 9.
$endgroup$
– userabc
Jan 4 at 5:02
$begingroup$
A slightly relevant remark, if there are four generators with similar relations, the group will be infinite. That is, $⟨a,b,c,d∣aba^{−1}=b^2, bcb^{−1}=c^2,cdc^{−1}=d^2, dad^{−1}=a^2⟩$ is not finite. c.f. Serre's Trees Page 9.
$endgroup$
– userabc
Jan 4 at 5:02
1
1
$begingroup$
@userabc Thanks for the comment - this is actually the very next remark in the paper I linked to. But the authors just call it a "well-known fact", so it's nice to have the reference to Serre's book. And I see that my question is Exercise 1) on p. 10 of Trees.
$endgroup$
– Alex Kruckman
Jan 4 at 5:14
$begingroup$
@userabc Thanks for the comment - this is actually the very next remark in the paper I linked to. But the authors just call it a "well-known fact", so it's nice to have the reference to Serre's book. And I see that my question is Exercise 1) on p. 10 of Trees.
$endgroup$
– Alex Kruckman
Jan 4 at 5:14
1
1
$begingroup$
... and now that I can search for an exercise number in a well-known book, I find that there are a number of questions on this site about the same group. For example, Jim Belk gave almost an identical proof to the one I found here, and Martin Brandenburg asked essentially the same question I'm asking here. Together, this evidence makes me think that there's unlikely to be a nicer proof. It would be reasonable to close this question as a duplicate.
$endgroup$
– Alex Kruckman
Jan 4 at 5:26
$begingroup$
... and now that I can search for an exercise number in a well-known book, I find that there are a number of questions on this site about the same group. For example, Jim Belk gave almost an identical proof to the one I found here, and Martin Brandenburg asked essentially the same question I'm asking here. Together, this evidence makes me think that there's unlikely to be a nicer proof. It would be reasonable to close this question as a duplicate.
$endgroup$
– Alex Kruckman
Jan 4 at 5:26
$begingroup$
There is another proof here.
$endgroup$
– Derek Holt
Jan 4 at 8:58
$begingroup$
There is another proof here.
$endgroup$
– Derek Holt
Jan 4 at 8:58
3
3
$begingroup$
Once you have proved that $a in langle b,c rangle$, you can use the more conceptual argument given by Bhaskar Vashishth in the linked proof, that $G$ is perfect, but $G = langle b,c rangle$ is solvable, so $G$ is trivial.
$endgroup$
– Derek Holt
Jan 4 at 15:05
$begingroup$
Once you have proved that $a in langle b,c rangle$, you can use the more conceptual argument given by Bhaskar Vashishth in the linked proof, that $G$ is perfect, but $G = langle b,c rangle$ is solvable, so $G$ is trivial.
$endgroup$
– Derek Holt
Jan 4 at 15:05
|
show 3 more comments
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5
$begingroup$
A slightly relevant remark, if there are four generators with similar relations, the group will be infinite. That is, $⟨a,b,c,d∣aba^{−1}=b^2, bcb^{−1}=c^2,cdc^{−1}=d^2, dad^{−1}=a^2⟩$ is not finite. c.f. Serre's Trees Page 9.
$endgroup$
– userabc
Jan 4 at 5:02
1
$begingroup$
@userabc Thanks for the comment - this is actually the very next remark in the paper I linked to. But the authors just call it a "well-known fact", so it's nice to have the reference to Serre's book. And I see that my question is Exercise 1) on p. 10 of Trees.
$endgroup$
– Alex Kruckman
Jan 4 at 5:14
1
$begingroup$
... and now that I can search for an exercise number in a well-known book, I find that there are a number of questions on this site about the same group. For example, Jim Belk gave almost an identical proof to the one I found here, and Martin Brandenburg asked essentially the same question I'm asking here. Together, this evidence makes me think that there's unlikely to be a nicer proof. It would be reasonable to close this question as a duplicate.
$endgroup$
– Alex Kruckman
Jan 4 at 5:26
$begingroup$
There is another proof here.
$endgroup$
– Derek Holt
Jan 4 at 8:58
3
$begingroup$
Once you have proved that $a in langle b,c rangle$, you can use the more conceptual argument given by Bhaskar Vashishth in the linked proof, that $G$ is perfect, but $G = langle b,c rangle$ is solvable, so $G$ is trivial.
$endgroup$
– Derek Holt
Jan 4 at 15:05