I'd just like to make sure I'm not messing up.
Take $A$ = $\begin{bmatrix}1 & 2 & 3 & 4 & 5\\2 & 1 & 3 & 5 & 4\end{bmatrix}$, and $B$ = $\begin{bmatrix}1 & 2 & 3 & 4 & 5\\5 & 4 & 1 & 2 & 3\end{bmatrix}$.
Then, AB gives us $\begin{bmatrix}1 & 2 & 3 & 4 & 5\\4 & 5 & 2 & 1 & 3\end{bmatrix}$.
How do I write these in cycle notation? I seem to be a bit confused. For instance, for $A$, 1 goes to 2 and 2 goes to 1. This would look like (1 2), but it neglects 3, 4, and 5. What am I missing? Thanks.
For $A$, once you get the disjoint cycle $(1 2)$ you move on to the other elements. So
$A=(1\ 2)(3)(4\ 5)=(1\ 2)(4\ 5)$.
It is optional to write any 1-cycle, so you may leave out $(3)$.
Similarly, $B=(1\ 5\ 3)(2\ 4)$.
$AB=(1\ 4)(2\ 5\ 3)$.
For $AB$, we go from left to right. For example, 1->5 in B, then 5->4 in A, so 1->4 in AB.