Let $sigma in S_n$ be a cycle. Prove that $sigma$ can be written as the product of at most $n-1$ transpositions. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Elements of $S_n$ which can not be product of $leq n-2$ transpositionsLogic for decomposing a permutation into different products composed of transpositionsShow that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.Odd PermutationsClarification on a cycle parity proofProve that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.Alternating Group ProofFinding the number of permutations in$S_9$ of the form $(a_1a_2)(a_3a_4)(a_5a_6)(a_7a_8a_9)$Prove that if $α ∈ S_n$ is an $m$-cycle, then $α$ is a product of transpositions.m-cycle as the product of distinct transpositionsLet $n$ be an odd positive integer and $ain S_n$ be an $n$-cycle. Show that the order of $C(a)$ must be odd.
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Let $sigma in S_n$ be a cycle. Prove that $sigma$ can be written as the product of at most $n-1$ transpositions.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Elements of $S_n$ which can not be product of $leq n-2$ transpositionsLogic for decomposing a permutation into different products composed of transpositionsShow that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.Odd PermutationsClarification on a cycle parity proofProve that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.Alternating Group ProofFinding the number of permutations in$S_9$ of the form $(a_1a_2)(a_3a_4)(a_5a_6)(a_7a_8a_9)$Prove that if $α ∈ S_n$ is an $m$-cycle, then $α$ is a product of transpositions.m-cycle as the product of distinct transpositionsLet $n$ be an odd positive integer and $ain S_n$ be an $n$-cycle. Show that the order of $C(a)$ must be odd.
$begingroup$
I know that $$(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_n-1)...(a_1a_3)(a_1a_2)$$
But how to prove this decomposition has the maximum number of non-repeating transpositions?
One start point may be that this decomposition contains all the symbols in the cycle, but then I have to prove all other decompositions that contain all the symbols in the cycle has no more transpositions than this one does, and I got stuck again.
I googled the question but to no avail.
Any help will be appreciated!
group-theory permutations permutation-cycles
edited Mar 27 at 9:04
Ernie060
2,940719
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
$endgroup$
add a comment |
$begingroup$
I know that $$(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_n-1)...(a_1a_3)(a_1a_2)$$
But how to prove this decomposition has the maximum number of non-repeating transpositions?
One start point may be that this decomposition contains all the symbols in the cycle, but then I have to prove all other decompositions that contain all the symbols in the cycle has no more transpositions than this one does, and I got stuck again.
I googled the question but to no avail.
Any help will be appreciated!
group-theory permutations permutation-cycles
edited Mar 27 at 9:04
Ernie060
2,940719
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
$endgroup$
4
$begingroup$
When you say "maximum", do you mean "minimum"?
$endgroup$
– Lord Shark the Unknown
Mar 27 at 5:49
$begingroup$
See also the answers to this question.
$endgroup$
– Dietrich Burde
Mar 27 at 10:02
add a comment |
$begingroup$
I know that $$(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_n-1)...(a_1a_3)(a_1a_2)$$
But how to prove this decomposition has the maximum number of non-repeating transpositions?
One start point may be that this decomposition contains all the symbols in the cycle, but then I have to prove all other decompositions that contain all the symbols in the cycle has no more transpositions than this one does, and I got stuck again.
I googled the question but to no avail.
Any help will be appreciated!
group-theory permutations permutation-cycles
edited Mar 27 at 9:04
Ernie060
2,940719
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
$endgroup$
I know that $$(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_n-1)...(a_1a_3)(a_1a_2)$$
But how to prove this decomposition has the maximum number of non-repeating transpositions?
One start point may be that this decomposition contains all the symbols in the cycle, but then I have to prove all other decompositions that contain all the symbols in the cycle has no more transpositions than this one does, and I got stuck again.
I googled the question but to no avail.
Any help will be appreciated!
group-theory permutations permutation-cycles
group-theory permutations permutation-cycles
edited Mar 27 at 9:04
Ernie060
2,940719
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
edited Mar 27 at 9:04
Ernie060
2,940719
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
edited Mar 27 at 9:04
Ernie060
2,940719
edited Mar 27 at 9:04
Ernie060
2,940719
edited Mar 27 at 9:04
Ernie060
2,940719
2,940719
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
asked Mar 27 at 5:36
Hua XiaoHua Xiao
191
191
4
$begingroup$
When you say "maximum", do you mean "minimum"?
$endgroup$
– Lord Shark the Unknown
Mar 27 at 5:49
$begingroup$
See also the answers to this question.
$endgroup$
– Dietrich Burde
Mar 27 at 10:02
add a comment |
4
$begingroup$
When you say "maximum", do you mean "minimum"?
$endgroup$
– Lord Shark the Unknown
Mar 27 at 5:49
$begingroup$
See also the answers to this question.
$endgroup$
– Dietrich Burde
Mar 27 at 10:02
4
4
$begingroup$
When you say "maximum", do you mean "minimum"?
$endgroup$
– Lord Shark the Unknown
Mar 27 at 5:49
$begingroup$
When you say "maximum", do you mean "minimum"?
$endgroup$
– Lord Shark the Unknown
Mar 27 at 5:49
$begingroup$
See also the answers to this question.
$endgroup$
– Dietrich Burde
Mar 27 at 10:02
$begingroup$
See also the answers to this question.
$endgroup$
– Dietrich Burde
Mar 27 at 10:02
add a comment |
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$begingroup$
When you say "maximum", do you mean "minimum"?
$endgroup$
– Lord Shark the Unknown
Mar 27 at 5:49
$begingroup$
See also the answers to this question.
$endgroup$
– Dietrich Burde
Mar 27 at 10:02