Permutation and cycles decomposition

Question

For the permutation cycle $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 3 & 1 & 5 & 2 \end{pmatrix}$, they decomposed it into $(3 4)(13)(45)(25)$. I don't see how this works because isn't the two-line notatoin basically saying
$1 \mapsto 4\mapsto 5\mapsto 2\mapsto 3\mapsto 1$ and the decomposition looks like it says: $3 \mapsto 4 \mapsto 1 \mapsto 3 \mapsto 4 \mapsto 5 \mapsto 2 \mapsto 5...$
Can anyone explain?

Answer 1

In the decomposition as a product of transpositions, the product corresponds to composition of bijections, starting from the right. Thus, for instance: $$2\mapsto 5\mapsto 4\mapsto 4\mapsto 3, \enspace\text{so that}\enspace 2\mapsto 3. $$