For two permutations $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)? \text{ (where }a,b,c,d \text{ are all distinct)}.$
I'm fairly new to group theory, so I apologize if this is an obvious result. Here's my attempt:
$\tau_1\tau_2=(a\space b)(c\space d)=\begin{bmatrix} a&b&c&d\\ b&a&c&d \end{bmatrix} \begin{bmatrix} a&b&c&d\\ a&b&d&c \end{bmatrix}=\begin{bmatrix} a&b&c&d&\\ b&a&d&c \end{bmatrix}$
but I'm not sure how to write this as $(d\space a\space c\space )(a\space b\space d)$.
Thanks.