[Math] Writing a Permutation as a product of Disjoint Cycles

Question

Write the following as a product of disjoint cycles:

$(1 3 2 5 6)(2 3)(4 6 5 1 2)$

I know from my solutions guide that the answer is:

$(1 2 4)(3 5)(6)$

but I don't know how to do that. I started by writing it as a product of transpositions as such:

$(1 3)(3 2)(2 5)(5 6)(2 3)(4 6)(6 5)(5 1)(1 2)$

I want to put this in standard form, but I don't know where to go from here…If anyone could help shed light on the procedure to do this, it would be very helpful.

Answer 1

First write this product with one permutation and then into product of disjoint cycles:

$$(1 3 6)(1357) (1234)= \left(\begin{matrix}1&2&3&4&5&6\\ 2&4&5&1&3&6\end{matrix}\right)=(1 2 4)(3 5).$$