Proving the order of quaternion group is 8The order of the Quaternion Group$G/Z$ cannot be isomorphic to quaternion groupQuestion about Quaternion group $Q_8$ and Dihedral group $D_8$Prove $Q_8$, the group generated by two complex matrices $A$ & $B$ (see below) is a nonabelian group of order 8.Proving that the binary operator of a group is commutative if the relation on the group is a partial orderevery subgroup of the quaternion group is normalProving that an element of a given group has an infinite order
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Proving the order of quaternion group is 8
The order of the Quaternion Group$G/Z$ cannot be isomorphic to quaternion groupQuestion about Quaternion group $Q_8$ and Dihedral group $D_8$Prove $Q_8$, the group generated by two complex matrices $A$ & $B$ (see below) is a nonabelian group of order 8.Proving that the binary operator of a group is commutative if the relation on the group is a partial orderevery subgroup of the quaternion group is normalProving that an element of a given group has an infinite order
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$begingroup$
Given this relation
$Q_8=left<i,jmid ij=j^-1i, ji=i^-1jright>$. I want to show $Q_8$ has order $8$.
I first tried to prove $i^4=1$, but I got stuck. Can anyone give me constructive hint or suggestion?
abstract-algebra group-theory quaternions group-presentation
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
asked 8 hours ago
aloeveraaloevera
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$endgroup$
add a comment
|
$begingroup$
Given this relation
$Q_8=left<i,jmid ij=j^-1i, ji=i^-1jright>$. I want to show $Q_8$ has order $8$.
I first tried to prove $i^4=1$, but I got stuck. Can anyone give me constructive hint or suggestion?
abstract-algebra group-theory quaternions group-presentation
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
$endgroup$
$begingroup$
You can always just take the Cayley table of $Q_8$ and verify that it satisfies the relations.
$endgroup$
– freakish
8 hours ago
add a comment
|
$begingroup$
Given this relation
$Q_8=left<i,jmid ij=j^-1i, ji=i^-1jright>$. I want to show $Q_8$ has order $8$.
I first tried to prove $i^4=1$, but I got stuck. Can anyone give me constructive hint or suggestion?
abstract-algebra group-theory quaternions group-presentation
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
$endgroup$
Given this relation
$Q_8=left<i,jmid ij=j^-1i, ji=i^-1jright>$. I want to show $Q_8$ has order $8$.
I first tried to prove $i^4=1$, but I got stuck. Can anyone give me constructive hint or suggestion?
abstract-algebra group-theory quaternions group-presentation
abstract-algebra group-theory quaternions group-presentation
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
edited 1 hour ago
Shaun
13.7k12 gold badges39 silver badges127 bronze badges
13.7k12 gold badges39 silver badges127 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
asked 8 hours ago
aloeveraaloevera
1115 bronze badges
1115 bronze badges
$begingroup$
You can always just take the Cayley table of $Q_8$ and verify that it satisfies the relations.
$endgroup$
– freakish
8 hours ago
add a comment
|
$begingroup$
You can always just take the Cayley table of $Q_8$ and verify that it satisfies the relations.
$endgroup$
– freakish
8 hours ago
$begingroup$
You can always just take the Cayley table of $Q_8$ and verify that it satisfies the relations.
$endgroup$
– freakish
8 hours ago
$begingroup$
You can always just take the Cayley table of $Q_8$ and verify that it satisfies the relations.
$endgroup$
– freakish
8 hours ago
add a comment
|
2 Answers
2
active
oldest
votes
$begingroup$
First, show $i^2=j^2$, then look at $i^4=ij^2i$
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
$endgroup$
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
add a comment
|
$begingroup$
It may be useful to try to break the group representation down further. Canonically, the term $j^-1i$ is represented as an element unique from $i$ and $j$. Let's call this element $k$. What do you notice about the products $ij$, $jk$ and $ki$? Moreover, what are $i^2$, $j^2$ and $k^2$, and do each of these constructions look familiar? That may help you in showing why $i^4=1$.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
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2 Answers
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active
oldest
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2 Answers
2
active
oldest
votes
active
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active
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votes
$begingroup$
First, show $i^2=j^2$, then look at $i^4=ij^2i$
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
$endgroup$
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
add a comment
|
$begingroup$
First, show $i^2=j^2$, then look at $i^4=ij^2i$
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
$endgroup$
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
add a comment
|
$begingroup$
First, show $i^2=j^2$, then look at $i^4=ij^2i$
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
$endgroup$
First, show $i^2=j^2$, then look at $i^4=ij^2i$
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
answered 8 hours ago
Empy2Empy2
35.7k1 gold badge28 silver badges64 bronze badges
35.7k1 gold badge28 silver badges64 bronze badges
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
add a comment
|
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
$begingroup$
I managed to show $i^2=j^2$ and $i^4=e$ with a help of your suggestion. But how do I show there are only $8$ elements?
$endgroup$
– aloevera
7 hours ago
add a comment
|
$begingroup$
It may be useful to try to break the group representation down further. Canonically, the term $j^-1i$ is represented as an element unique from $i$ and $j$. Let's call this element $k$. What do you notice about the products $ij$, $jk$ and $ki$? Moreover, what are $i^2$, $j^2$ and $k^2$, and do each of these constructions look familiar? That may help you in showing why $i^4=1$.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment
|
$begingroup$
It may be useful to try to break the group representation down further. Canonically, the term $j^-1i$ is represented as an element unique from $i$ and $j$. Let's call this element $k$. What do you notice about the products $ij$, $jk$ and $ki$? Moreover, what are $i^2$, $j^2$ and $k^2$, and do each of these constructions look familiar? That may help you in showing why $i^4=1$.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment
|
$begingroup$
It may be useful to try to break the group representation down further. Canonically, the term $j^-1i$ is represented as an element unique from $i$ and $j$. Let's call this element $k$. What do you notice about the products $ij$, $jk$ and $ki$? Moreover, what are $i^2$, $j^2$ and $k^2$, and do each of these constructions look familiar? That may help you in showing why $i^4=1$.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
It may be useful to try to break the group representation down further. Canonically, the term $j^-1i$ is represented as an element unique from $i$ and $j$. Let's call this element $k$. What do you notice about the products $ij$, $jk$ and $ki$? Moreover, what are $i^2$, $j^2$ and $k^2$, and do each of these constructions look familiar? That may help you in showing why $i^4=1$.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
answered 7 hours ago
JJ HooJJ Hoo
662 bronze badges
662 bronze badges
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
JJ Hoo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment
|
add a comment
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$begingroup$
You can always just take the Cayley table of $Q_8$ and verify that it satisfies the relations.
$endgroup$
– freakish
8 hours ago