Consider the permutation $(2,6,3,4,5) \in S_6$. Then it's transposition form would be $(2,5), (2,4), (2,3), (2,6)$. But I don't understand how to show that $(2,6,3,4,5) \in S_6$ can be written as a product of simple transpositions. I know a transposition, $\sigma \in S_n$, is simple if it is of the form $\sigma = (i, i+1)$ for some $i \in \left \{ 1,2,...,n-1 \right \}$. I've looked at other examples, but I still don't think I understand. Below is what I've tried, but I really don't think this is right. How am I suppose to do this?
Given $(2,6,3,4,5) \in S_6$ which can be written as $(2,5), (2,4), (2,3), (2,6)$. Then
$(2,5) = (4,5), (3,4), (2,3), (3,4), (4,5)$
$(2,4) = (3,4), (2,3), (3,4) $
$(2,6) = (5,6),(4,5), (3,4), (2,3), (4,5), (5,6)$
Therefore, $(2,6,3,4,5) \in S_6= (4,5), (3,4), (2,3), (3,4), (4,5)(3,4), (2,3), (3,4)(2,3)(5,6),(4,5), (3,4), (2,3), (4,5), (5,6)$