How to write permutations as product of disjoint cycles and transpositions

I'll use a longer cycle to help describe two techniques for writing disjoint cycles as the product of transpositions:

Let's say $\tau = (1, 3, 4, 6, 7, 9) \in S_9$

Then, note the patterns:

Method 1: $\tau = (1, 3, 4, 6, 7, 9) = (1, 9)(1, 7)(1, 6)(1, 4)(1, 3)$

Method 2: $\tau = (1, 3, 4, 6, 7, 9) = (1, 3)(3, 4)(4, 6)(6, 7)(7, 9)$

Both products of transpositions, method $1$ or method $2$, represent the same permutation, $\tau$. Note that the order of the disjoint cycle $\tau$ is $6$, but in both expressions of $\tau$ as the product of transpositions, $\tau$ has $5$ (odd number of) transpositions. Hence $\tau$ is an odd permutation.

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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 $\sigma = \operatorname{lcm}(3, 5, 2) = 30$.

• Expressing $\sigma $ as the product of transpositions:

  • $\sigma =(1, 4)(4, 10)(3, 9)(9, 8)(8, 7)(7, 11)(5, 6):\quad 7$ transpositions in all, so $\sigma $ is an odd permutation (which happens to be of even order).