permutations as product of disjoint cycles I need help understanding. So here is a sample question
write the following permutations as a product of disjoint cycles
(1235)(413) it was given that the disjoint cycles is (15) (234)
could some one help me understand the steps of how to get to this point please
 A: This is a quite simple problem, once you get the hang of it.
We simply look at the first element that we would like to write down, which is typically going to be 1. Note that I am strictly looking at (1235)(413) right now. I am looking at this because we see that this is not disjoint, as there are elements in (1235) that are also in (413). To be disjoint, we should have all of the elements we want on one side, such as the finished for of this example. So, let's get started!
We first take 1 (reading from right to left), and we see that 1 goes to 3. Now, we go to the next cycle, (1235), and we see that 3 goes to 5. Thus, 1 goes to five. The way I was taught from here may be different, but it is typically easier to now look at five in our question.
We see that 5 is not in the first cycle, (413), so we look at the next cycle, and see that 5 goes to 1. Thus, we have a completed cycle! That is, (1 5) is the completed cycle. We then can repeat this and see the following:
 3 --> 4 --> 4
 4 --> 1 --> 2
 2 --> 2 --> 3

Thus, we see that the rest of the elements can be written in there own cycle, (2 3 4). And we have our final answer to be:
 (1 5)(2 3 4).

If you have any more questions, feel free to ask!
A: First we write the given product of cycles on one permutation then we decompose it on a disjoint cycles:
$$(1235)(413)=\left(\begin{matrix}1&2&3&4&5\\
5&3&4&2&1\end{matrix}\right)=(15)(234)$$
