Regular permutations
$begingroup$
$ sigma in S_n $ is a regular permutation if there exist disjoint cycles $ tau_1...tau_r $ of length m such that $ sigma = tau_1 circ ... circ tau_r $ and $supp(tau_1)cup...cup supp(tau_r) = {1...n}$
I have been asked to prove that:
$sigma $ is a regular permutation $Longleftrightarrow$ $ exists $ some $k$ natural and some cycle $tau in S_n $ such that $sigma = tau^k$.
I have observed that $ n =rm$ , and that the orbit of $sigma$ is ${1...n}$ , but I just cannot figure out how to continue. Can someone give me some tips?
Thank you in advance.
group-theory permutations
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
$endgroup$
add a comment |
$begingroup$
$ sigma in S_n $ is a regular permutation if there exist disjoint cycles $ tau_1...tau_r $ of length m such that $ sigma = tau_1 circ ... circ tau_r $ and $supp(tau_1)cup...cup supp(tau_r) = {1...n}$
I have been asked to prove that:
$sigma $ is a regular permutation $Longleftrightarrow$ $ exists $ some $k$ natural and some cycle $tau in S_n $ such that $sigma = tau^k$.
I have observed that $ n =rm$ , and that the orbit of $sigma$ is ${1...n}$ , but I just cannot figure out how to continue. Can someone give me some tips?
Thank you in advance.
group-theory permutations
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
$endgroup$
$begingroup$
There seems to be some kind of error in the statement. As written, it is obviously false since you could always pick $tau=sigma$ and $k=1$.
$endgroup$
– Eric Wofsey
Jan 23 at 20:36
$begingroup$
Perhaps $sigma$ is supposed to be an $n$-cycle and $k=r$
$endgroup$
– Derek Holt
Jan 23 at 20:45
$begingroup$
Yes , I forgot the cycle part , I have just corrected it , sorry.
$endgroup$
– Adolfo Garcia
Jan 23 at 21:17
add a comment |
$begingroup$
$ sigma in S_n $ is a regular permutation if there exist disjoint cycles $ tau_1...tau_r $ of length m such that $ sigma = tau_1 circ ... circ tau_r $ and $supp(tau_1)cup...cup supp(tau_r) = {1...n}$
I have been asked to prove that:
$sigma $ is a regular permutation $Longleftrightarrow$ $ exists $ some $k$ natural and some cycle $tau in S_n $ such that $sigma = tau^k$.
I have observed that $ n =rm$ , and that the orbit of $sigma$ is ${1...n}$ , but I just cannot figure out how to continue. Can someone give me some tips?
Thank you in advance.
group-theory permutations
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
$endgroup$
$ sigma in S_n $ is a regular permutation if there exist disjoint cycles $ tau_1...tau_r $ of length m such that $ sigma = tau_1 circ ... circ tau_r $ and $supp(tau_1)cup...cup supp(tau_r) = {1...n}$
I have been asked to prove that:
$sigma $ is a regular permutation $Longleftrightarrow$ $ exists $ some $k$ natural and some cycle $tau in S_n $ such that $sigma = tau^k$.
I have observed that $ n =rm$ , and that the orbit of $sigma$ is ${1...n}$ , but I just cannot figure out how to continue. Can someone give me some tips?
Thank you in advance.
group-theory permutations
group-theory permutations
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
edited Jan 23 at 21:17
Adolfo Garcia
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
asked Jan 23 at 19:43
Adolfo GarciaAdolfo Garcia
142
142
$begingroup$
There seems to be some kind of error in the statement. As written, it is obviously false since you could always pick $tau=sigma$ and $k=1$.
$endgroup$
– Eric Wofsey
Jan 23 at 20:36
$begingroup$
Perhaps $sigma$ is supposed to be an $n$-cycle and $k=r$
$endgroup$
– Derek Holt
Jan 23 at 20:45
$begingroup$
Yes , I forgot the cycle part , I have just corrected it , sorry.
$endgroup$
– Adolfo Garcia
Jan 23 at 21:17
add a comment |
$begingroup$
There seems to be some kind of error in the statement. As written, it is obviously false since you could always pick $tau=sigma$ and $k=1$.
$endgroup$
– Eric Wofsey
Jan 23 at 20:36
$begingroup$
Perhaps $sigma$ is supposed to be an $n$-cycle and $k=r$
$endgroup$
– Derek Holt
Jan 23 at 20:45
$begingroup$
Yes , I forgot the cycle part , I have just corrected it , sorry.
$endgroup$
– Adolfo Garcia
Jan 23 at 21:17
$begingroup$
There seems to be some kind of error in the statement. As written, it is obviously false since you could always pick $tau=sigma$ and $k=1$.
$endgroup$
– Eric Wofsey
Jan 23 at 20:36
$begingroup$
There seems to be some kind of error in the statement. As written, it is obviously false since you could always pick $tau=sigma$ and $k=1$.
$endgroup$
– Eric Wofsey
Jan 23 at 20:36
$begingroup$
Perhaps $sigma$ is supposed to be an $n$-cycle and $k=r$
$endgroup$
– Derek Holt
Jan 23 at 20:45
$begingroup$
Perhaps $sigma$ is supposed to be an $n$-cycle and $k=r$
$endgroup$
– Derek Holt
Jan 23 at 20:45
$begingroup$
Yes , I forgot the cycle part , I have just corrected it , sorry.
$endgroup$
– Adolfo Garcia
Jan 23 at 21:17
$begingroup$
Yes , I forgot the cycle part , I have just corrected it , sorry.
$endgroup$
– Adolfo Garcia
Jan 23 at 21:17
add a comment |
1 Answer
1
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$begingroup$
Interleave the entries from each cycle to create one big cycle that starts with the first element of each cycle then (in the same ordering of cycles) the second element of each cycle, then the third element of each cycle, etc.
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
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add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
votes
$begingroup$
Interleave the entries from each cycle to create one big cycle that starts with the first element of each cycle then (in the same ordering of cycles) the second element of each cycle, then the third element of each cycle, etc.
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
$endgroup$
add a comment |
$begingroup$
Interleave the entries from each cycle to create one big cycle that starts with the first element of each cycle then (in the same ordering of cycles) the second element of each cycle, then the third element of each cycle, etc.
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
$endgroup$
add a comment |
$begingroup$
Interleave the entries from each cycle to create one big cycle that starts with the first element of each cycle then (in the same ordering of cycles) the second element of each cycle, then the third element of each cycle, etc.
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
$endgroup$
Interleave the entries from each cycle to create one big cycle that starts with the first element of each cycle then (in the same ordering of cycles) the second element of each cycle, then the third element of each cycle, etc.
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
answered Jan 24 at 1:53
C MonsourC Monsour
6,2541325
6,2541325
add a comment |
add a comment |
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$begingroup$
There seems to be some kind of error in the statement. As written, it is obviously false since you could always pick $tau=sigma$ and $k=1$.
$endgroup$
– Eric Wofsey
Jan 23 at 20:36
$begingroup$
Perhaps $sigma$ is supposed to be an $n$-cycle and $k=r$
$endgroup$
– Derek Holt
Jan 23 at 20:45
$begingroup$
Yes , I forgot the cycle part , I have just corrected it , sorry.
$endgroup$
– Adolfo Garcia
Jan 23 at 21:17