<s>[INST] Permutation $\alpha^2$ Show that there is no permutation $\alpha \in S_4$ so that $\alpha^2=\begin{pmatrix}
1 &3  &4  &2 
\end{pmatrix}$.
$S_4=\begin{pmatrix}
1 &2 &3  &4 
\end{pmatrix}$
If $ \alpha=\begin{pmatrix}
2 &4  &3
\end{pmatrix}$, we get:
$$\alpha^2=\alpha\cdot\alpha=\begin{pmatrix}
1 &2  &3  &4 \\ 
 1&4  &2  &3 \\1 
 &3  &4  &2 
\end{pmatrix}$$
Why can't I do that?
 [/INST] $\beta = (1 \ 3 \ 4 \ 2)$ is on odd permutation: a cycle of length equal to $4$. As the square of any permutation is even, $\beta$ can't be written as a square.
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