# Product of disjoint cycles and product of transpositions

$\alpha=(3412)(245)\in S_5$ and I have to 1) write this as a product of disjoint cycles, 2) write this as a product of transpositions.

1) I can do thing by following where the elements go in the two permutations. I obtain $\alpha=(21)(453)$ this seems to be correct.

2) How do I write this as a product of transpostions? I have trouble seeing how this is possible, seeing that $\alpha$ has a three length cycle within it.

How do I do 2)?

• $\alpha=(2\ 1)(4\ 5)(5\ 3)$ – bof Jun 19 '15 at 2:03
• @bof I'm an idiot Edit: thanks! – Permute Jun 19 '15 at 2:04

Once a permutation is known as a product of disjoint (hence commuting) cycles, one can work in parallel and so it suffices to know how to write a cycle as a product of transpositions. To avoid a clutter of subscripts, I'll give an example:

$(7413625) = (75)(72)(76)(73)(71)(74)$

(In each of these transpositions 7 occurs, the other numbers accompanying 7 are taken in the reverse order to the given cycle)

• That's highly convenient, thanks! – Permute Jun 19 '15 at 2:23
• So there are multiple products of transpositions? Doing the product of transpostions off of $(3412)(245)$ gives us $(32)(31)(34)(25)(24)$ which works and doing it off of $(21)(453)$ gives us $(21)(43)(45)$ which also works. – Permute Jun 19 '15 at 2:38
• @Permute Indeed. Notice, however, that you can only write $\alpha$ as a product of an odd number of transpositions. – Taylor Jun 19 '15 at 2:43
• @Taylor Does such a property imply $\alpha$ is odd? That was actually another part of the past exam question. – Permute Jun 19 '15 at 2:56
• @Taylor Sorry I should restate my question, does $\alpha$ being odd or even, mean that it has an odd or even number of transpositions in its product of transpositions. Or does it refer to something else? – Permute Jun 19 '15 at 3:10

As Product of disjoint cycles: $\alpha=(3412)(245)=(12)(345)$

As Product of transpositions: $\alpha=(3412)(245)=(32)(31)(34)(25)(24)$