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## How to write a permutation as a product of transpositions

Number of Transpositions in a Permutation - GeeksforGeeks Ways of expressing permutations as products of transpositions abstract algebra - How to write permutations as product of disjoint Number of Transpositions in a Permutation - GeeksforGeeks Method 2: τ = ( 1, 3, 4, 6, 7, 9) = ( 1, 3) ( 3, 4) ( 4, 6) ( 6, 7) ( 7, 9) Both products of transpositions, method 1 or method 2, represent the same permutation, τ. Note that the order of the disjoint cycle τ is 6, but in both expressions of τ as the product of. To see why they are equal, you might find it helpful to first write your permutation as a product of disjoint cycles. So, for example, assuming you are composing products of permutation from left to right, then note that $\varphi = (1527)(3567)(273)$ is not a product of disjoint cycles: we see that $7$ appears in all three of the factors, $2$ , $3$ and $5$ each appear twice, etc. Expressing permutation as a product of Transpositions. Expressing permutation as a product of Transpositions and examples on it. Now, don't forget to multiply the transpositions you obtain for each disjoint cycle so you obtain an expression of the permutation $S_ {11}$ as the product of the product of transpositions, and determine whether it is odd or even: $\sigma = (1, 4, 10) (3, 9, 8, 7, 11) (5, 6)$. The order of $\sigma = \operatorname {lcm} (3, 5, 2) = 30$. Theorem 1.

A permutation in Sn cannot be written as the product of fewer than n - r transpositions, where r is the number of disjoint cycles in the permutation. Proof: Suppose ff in Sn is written as ff = 7172 * * * Tk where the ri's are transposi-tions. Viewing transpositions as reflections through hyperplanes, let A permutation cycle is a set of elements in a permutation that trade places with one another. For e.g. P = { 5, 1, 4, 2, 3 }: Here, 5 goes to 1, 1 goes to 2 and so on (according to their indices position): 5 -> 1 1 -> 2 2 -> 4 4 -> 3 3 -> 5 Thus it can be represented as a single cycle: (5, 1, 2, 4, 3). Now consider the permutation: {5, 1, 4, 3, 2}. We give two examples of writing a permutation written as a product of nondisjoint cycles as a product of disjoint cycles (with one factor).

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