Let $G$ be a group and suppose that $a,bin G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove that $b^5=1$. [closed]
$begingroup$
I recently found an exercise about group presentation that I have no idea how to work out.
Can anyone help me?
Let $G$ be a group and suppose that $a,bin G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove that $b^5=1$.
group-theory group-presentation
edited Dec 10 '18 at 6:07
Shaun
8,832113681
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
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closed as off-topic by Shaun, Derek Holt, Gibbs, John B, José Carlos Santos Dec 10 '18 at 11:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shaun, Derek Holt, Gibbs, John B, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I recently found an exercise about group presentation that I have no idea how to work out.
Can anyone help me?
Let $G$ be a group and suppose that $a,bin G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove that $b^5=1$.
group-theory group-presentation
edited Dec 10 '18 at 6:07
Shaun
8,832113681
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
$endgroup$
closed as off-topic by Shaun, Derek Holt, Gibbs, John B, José Carlos Santos Dec 10 '18 at 11:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shaun, Derek Holt, Gibbs, John B, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I recently found an exercise about group presentation that I have no idea how to work out.
Can anyone help me?
Let $G$ be a group and suppose that $a,bin G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove that $b^5=1$.
group-theory group-presentation
edited Dec 10 '18 at 6:07
Shaun
8,832113681
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
$endgroup$
I recently found an exercise about group presentation that I have no idea how to work out.
Can anyone help me?
Let $G$ be a group and suppose that $a,bin G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove that $b^5=1$.
group-theory group-presentation
group-theory group-presentation
edited Dec 10 '18 at 6:07
Shaun
8,832113681
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
edited Dec 10 '18 at 6:07
Shaun
8,832113681
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
edited Dec 10 '18 at 6:07
Shaun
8,832113681
edited Dec 10 '18 at 6:07
Shaun
8,832113681
edited Dec 10 '18 at 6:07
Shaun
8,832113681
8,832113681
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
asked Dec 5 '18 at 8:38
Censi LICensi LI
3,6161937
3,6161937
closed as off-topic by Shaun, Derek Holt, Gibbs, John B, José Carlos Santos Dec 10 '18 at 11:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shaun, Derek Holt, Gibbs, John B, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Shaun, Derek Holt, Gibbs, John B, José Carlos Santos Dec 10 '18 at 11:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shaun, Derek Holt, Gibbs, John B, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
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Since $a^2=1$, that is, $a=a^{-1}$, the relation $ab^2a=b^3$ implies $b^2a=ab^3$ and $ab^2=b^3a$. These then imply $b^2a=ab^3=b^3ab$, hence $a=bab$. Then $1=a^2=(bab)^2=dots$
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Since $a^2=1$, that is, $a=a^{-1}$, the relation $ab^2a=b^3$ implies $b^2a=ab^3$ and $ab^2=b^3a$. These then imply $b^2a=ab^3=b^3ab$, hence $a=bab$. Then $1=a^2=(bab)^2=dots$
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
$endgroup$
add a comment |
$begingroup$
Since $a^2=1$, that is, $a=a^{-1}$, the relation $ab^2a=b^3$ implies $b^2a=ab^3$ and $ab^2=b^3a$. These then imply $b^2a=ab^3=b^3ab$, hence $a=bab$. Then $1=a^2=(bab)^2=dots$
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
$endgroup$
add a comment |
$begingroup$
Since $a^2=1$, that is, $a=a^{-1}$, the relation $ab^2a=b^3$ implies $b^2a=ab^3$ and $ab^2=b^3a$. These then imply $b^2a=ab^3=b^3ab$, hence $a=bab$. Then $1=a^2=(bab)^2=dots$
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
$endgroup$
Since $a^2=1$, that is, $a=a^{-1}$, the relation $ab^2a=b^3$ implies $b^2a=ab^3$ and $ab^2=b^3a$. These then imply $b^2a=ab^3=b^3ab$, hence $a=bab$. Then $1=a^2=(bab)^2=dots$
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
answered Dec 5 '18 at 8:54
BWWBWW
9,25622237
9,25622237
add a comment |
add a comment |
