How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$ I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that
$$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$
I started following the path of $2$ on the right:
$$2\to3\to4\to5\to \ ?$$
Where does $5$ go? I should stop here, right? Then $2\to 5$, that is, $(25)$. But what about $(23)(34)$?

Let's go through the product(s) 
$$
(2 3) (3 4) (4 5) (4 3) (3 2)
$$
step by step.
Start with $2$, by writing $(2$, and see how the elements map out:


*

*$2\mapsto 3\mapsto 4\mapsto 5\mapsto 5\mapsto 5\qquad:\qquad(2\; 5$

*$5\mapsto 5\mapsto 5\mapsto 4\mapsto 3\mapsto 2\qquad:\qquad(2\; 5)(3$

*$3\mapsto 2\mapsto 2\mapsto 2\mapsto 2\mapsto 3\qquad:\qquad(2\; 5)(3)(4$

*$4\mapsto 4\mapsto 3\mapsto 3\mapsto 4\mapsto 4\qquad:\qquad(2\;5)(3)(4)$


The process has terminated and we know $(2\;5)(3)(4)=(2\;5)$. That is, we can see that
$$
(2\;5) = (2 3) (3 4) (4 5) (4 3) (3 2).
$$
Did all of those steps make sense?
