Source Page No $1$.
Suppose that $G = S_4$, the group of permutations on the set $S = \{1, 2, 3, 4\}$. We illustrate the action of $G$ on $S$ as in the following examples:
$$(12)(34).3 = 4.$$
My confusion: why is $$(12)(34).3 = 4?$$
My thinking: Here $(12)(34).(3)=\begin{pmatrix} 1&2&3&4 \\2 &1&4&3 \end{pmatrix}(3)=(12)(34)$
Maybe I'm wrong.
Here $3$ is only one cycle. How can we apply the product of permutation?
Note: I know the rule of product of permutation for example I can calculate $(1324)(243)=142$
Yes,
$$\begin{align} (12)(34).3&=(12)(34)3\\ &=4. \end{align}$$
This can be seen by splitting $(12)(34)$ into its two components: $f:=(12)$ and $g:=(34)$. Then
$$3\xrightarrow{g}4\xrightarrow{f}4.$$
Your confusion is in thinking $3$ is a one-cycle; rather, it is an argument of the function $f\circ g=(12)(34)$.
By the way,
$$(1324)(243)=(1342).$$