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 $(23)(34)$?

Assuming there are cycles in $S_5$, $$(2\ 3) = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 2 & 4 & 5 \end{pmatrix}=(2\ 3)(1)(4)(5) \\
(3\ 4) = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 4 & 3 & 5 \end{pmatrix}=(3\ 4)(1)(2)(5)$$
Hence, $5\mapsto 5$ in both of those cycles.
However, to conclude that $5 \mapsto 2$ you need to follow it through the cycles as you followed 2, starting from the right to the left.
See my answer here for a detailed explanation of composing cycles: Shortcut for composing cycles
