cardinal of conjugacy class of $(123)(456)$ in $A_7$ [duplicate]
This question already has an answer here:
Splitting of conjugacy class in alternating group
1 answer
So it is not hard to see that the conjugacy class of $(123)(456)$ in $S_7$ has cardinality $frac{7!}{3cdot 3cdot2}=280$.
But for instance to go from $(123)(456)$ to $(132)(456)$ you have to conjugate by $(23)$ which is not in $A_7$ so $(132)(456)$ is not in our class (call it $C$). Same for $(123)(465)$. But $(132)(465)in C$ because $(23)(56)in A_7$.
I could probably find all the elements of $C$ by trying out different permutations but I feel like there is a better way...
We know that $|A_7|={|S_7|over2}$ and conjugating $(123)(456)$ by any permutation in $S_7supset A_7$ gives a permutation of the type $(abc)(def)~$ with $a,...,fin{1,...,7}$ and $a,b,...,f$ are disjoint because otherwise the "permutation" by which we conjugate wouldn't be bijective.
So I'm tempted to say that $|C|=|bigcuplimits_{~ain A_7}{(~a(1)a(2)a(3)~)~(~a(4)a(5)a(6)~)~}|=frac{280}{2}$ but that's probably false.
A hint would be appreciated
group-theory finite-groups symmetric-groups
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
marked as duplicate by Jyrki Lahtonen group-theory
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2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
This question already has an answer here:
Splitting of conjugacy class in alternating group
1 answer
So it is not hard to see that the conjugacy class of $(123)(456)$ in $S_7$ has cardinality $frac{7!}{3cdot 3cdot2}=280$.
But for instance to go from $(123)(456)$ to $(132)(456)$ you have to conjugate by $(23)$ which is not in $A_7$ so $(132)(456)$ is not in our class (call it $C$). Same for $(123)(465)$. But $(132)(465)in C$ because $(23)(56)in A_7$.
I could probably find all the elements of $C$ by trying out different permutations but I feel like there is a better way...
We know that $|A_7|={|S_7|over2}$ and conjugating $(123)(456)$ by any permutation in $S_7supset A_7$ gives a permutation of the type $(abc)(def)~$ with $a,...,fin{1,...,7}$ and $a,b,...,f$ are disjoint because otherwise the "permutation" by which we conjugate wouldn't be bijective.
So I'm tempted to say that $|C|=|bigcuplimits_{~ain A_7}{(~a(1)a(2)a(3)~)~(~a(4)a(5)a(6)~)~}|=frac{280}{2}$ but that's probably false.
A hint would be appreciated
group-theory finite-groups symmetric-groups
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
marked as duplicate by Jyrki Lahtonen group-theory
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2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Hint: Can you show that if $C$ is a conjugacy class in $S_{n}$ that is a subset of $A_{n}$, then it is either a conjugacy class in $A_{n}$ or it splits into a union of two $A_{n}$ conjugacy classes.
– Adam Higgins
Jan 6 at 11:27
add a comment |
This question already has an answer here:
Splitting of conjugacy class in alternating group
1 answer
So it is not hard to see that the conjugacy class of $(123)(456)$ in $S_7$ has cardinality $frac{7!}{3cdot 3cdot2}=280$.
But for instance to go from $(123)(456)$ to $(132)(456)$ you have to conjugate by $(23)$ which is not in $A_7$ so $(132)(456)$ is not in our class (call it $C$). Same for $(123)(465)$. But $(132)(465)in C$ because $(23)(56)in A_7$.
I could probably find all the elements of $C$ by trying out different permutations but I feel like there is a better way...
We know that $|A_7|={|S_7|over2}$ and conjugating $(123)(456)$ by any permutation in $S_7supset A_7$ gives a permutation of the type $(abc)(def)~$ with $a,...,fin{1,...,7}$ and $a,b,...,f$ are disjoint because otherwise the "permutation" by which we conjugate wouldn't be bijective.
So I'm tempted to say that $|C|=|bigcuplimits_{~ain A_7}{(~a(1)a(2)a(3)~)~(~a(4)a(5)a(6)~)~}|=frac{280}{2}$ but that's probably false.
A hint would be appreciated
group-theory finite-groups symmetric-groups
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
This question already has an answer here:
Splitting of conjugacy class in alternating group
1 answer
So it is not hard to see that the conjugacy class of $(123)(456)$ in $S_7$ has cardinality $frac{7!}{3cdot 3cdot2}=280$.
But for instance to go from $(123)(456)$ to $(132)(456)$ you have to conjugate by $(23)$ which is not in $A_7$ so $(132)(456)$ is not in our class (call it $C$). Same for $(123)(465)$. But $(132)(465)in C$ because $(23)(56)in A_7$.
I could probably find all the elements of $C$ by trying out different permutations but I feel like there is a better way...
We know that $|A_7|={|S_7|over2}$ and conjugating $(123)(456)$ by any permutation in $S_7supset A_7$ gives a permutation of the type $(abc)(def)~$ with $a,...,fin{1,...,7}$ and $a,b,...,f$ are disjoint because otherwise the "permutation" by which we conjugate wouldn't be bijective.
So I'm tempted to say that $|C|=|bigcuplimits_{~ain A_7}{(~a(1)a(2)a(3)~)~(~a(4)a(5)a(6)~)~}|=frac{280}{2}$ but that's probably false.
A hint would be appreciated
This question already has an answer here:
Splitting of conjugacy class in alternating group
1 answer
group-theory finite-groups symmetric-groups
group-theory finite-groups symmetric-groups
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
asked Jan 6 at 11:03
John CataldoJohn Cataldo
1,0961216
1,0961216
marked as duplicate by Jyrki Lahtonen group-theory
Users with the group-theory badge can single-handedly close group-theory questions as duplicates and reopen them as needed.
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2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Jyrki Lahtonen group-theory
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2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Hint: Can you show that if $C$ is a conjugacy class in $S_{n}$ that is a subset of $A_{n}$, then it is either a conjugacy class in $A_{n}$ or it splits into a union of two $A_{n}$ conjugacy classes.
– Adam Higgins
Jan 6 at 11:27
add a comment |
Hint: Can you show that if $C$ is a conjugacy class in $S_{n}$ that is a subset of $A_{n}$, then it is either a conjugacy class in $A_{n}$ or it splits into a union of two $A_{n}$ conjugacy classes.
– Adam Higgins
Jan 6 at 11:27
Hint: Can you show that if $C$ is a conjugacy class in $S_{n}$ that is a subset of $A_{n}$, then it is either a conjugacy class in $A_{n}$ or it splits into a union of two $A_{n}$ conjugacy classes.
– Adam Higgins
Jan 6 at 11:27
Hint: Can you show that if $C$ is a conjugacy class in $S_{n}$ that is a subset of $A_{n}$, then it is either a conjugacy class in $A_{n}$ or it splits into a union of two $A_{n}$ conjugacy classes.
– Adam Higgins
Jan 6 at 11:27
add a comment |
1 Answer
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Hint: If you conjugate $(123)(456)$ by $(14)(25)(36)$, an odd permutation, you get $(123)(456)$ back. Therefore you also get $(132)(456)$ from $(123)(456)$ by conjugating it by an even permutation.
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint: If you conjugate $(123)(456)$ by $(14)(25)(36)$, an odd permutation, you get $(123)(456)$ back. Therefore you also get $(132)(456)$ from $(123)(456)$ by conjugating it by an even permutation.
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
add a comment |
Hint: If you conjugate $(123)(456)$ by $(14)(25)(36)$, an odd permutation, you get $(123)(456)$ back. Therefore you also get $(132)(456)$ from $(123)(456)$ by conjugating it by an even permutation.
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
add a comment |
Hint: If you conjugate $(123)(456)$ by $(14)(25)(36)$, an odd permutation, you get $(123)(456)$ back. Therefore you also get $(132)(456)$ from $(123)(456)$ by conjugating it by an even permutation.
Hint: If you conjugate $(123)(456)$ by $(14)(25)(36)$, an odd permutation, you get $(123)(456)$ back. Therefore you also get $(132)(456)$ from $(123)(456)$ by conjugating it by an even permutation.
edited Jan 6 at 11:30
community wiki
2 revs
Jyrki Lahtonen
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
add a comment |
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
This theme comes up frequently on our site, hence only a hint in a community wiki post. This if probably one of the highest voted threads getting to, if not the bottom of the matter, then at least relatively close. There are many other good duplicate targets.
– Jyrki Lahtonen
Jan 6 at 11:27
add a comment |
Hint: Can you show that if $C$ is a conjugacy class in $S_{n}$ that is a subset of $A_{n}$, then it is either a conjugacy class in $A_{n}$ or it splits into a union of two $A_{n}$ conjugacy classes.
– Adam Higgins
Jan 6 at 11:27