How many permutations $\theta$ on $\{1,2,3,4\}$ have $\theta(4)=4$.
Request vetting
In permutations, always have bijective mappings.
After fixing one element have options available:
$=3,$ for first element,
$=2,$ for second element,
$=1,$ for third element.
Say, now elements left are three: $\{1,2,3\}$, the mappings possible are : $$ \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1\end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3\end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1\end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3\end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2\end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2\end{pmatrix} $$
Want to add another question to it, for knowledge enhancement. Find conjugate class elements of each of the six permutations above.
Request some hints, and if possible an example too. That would assist in handling bigger examples in future.
Edit
Based on response by @StinkingBishop, get: Two permutations are conjugate iff they have the same lengths of cycles, regardless of which elements belong to the cycles.
There are three cycle lengths, each forms a conjugacy class with order (= number of elements in conjugacy class).
$1: \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3\end{pmatrix}$
$2: \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1\end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3\end{pmatrix},
\begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2\end{pmatrix}
$
$3: \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1\end{pmatrix}
\begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2\end{pmatrix}
$