Let $\begin{equation*} p = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 8 & 4 & 7 & 3 & 6 & 2 & 5 \\ \end{pmatrix} \end{equation*}$
I would like to write $p$ as a product of transpositions. To do this, I first write out $p$ as a product of disjoint cycles as follows:
$p = (6)\circ(2, 8, 5, 3, 4, 7)\circ(1)$
or, in simpler terms, given the fact that we can drop all single element brackets:
$p = (2, 8, 5, 3, 4, 7)$
I know this is correct because when I calculate the above permutation, I end up with the given $p$.
My challenge now comes in when I expand on this intermediary step and write out the product of transpositions, i.e. splitting these cycles to length 2. According to @amWhy's answer on this thread, I use the quoted method with an example given: $Method 1: τ=(1,3,4,6,7,9)=(1,9)(1,7)(1,6)(1,4)(1,3)$
Using this method on my example gives this answer: $p = (2,7)\circ(2,4)\circ(2,3)\circ(2,5)\circ(2,8)$
However checking this answer by multiplying out this product doesnt give me the original $p$ in the question.
Where did I go wrong?
