<s>[INST] Permutation's decomposition into transpositions 
*

*Transposition is a cycle with 2 elements.

*Any permutation can be decomposed into a product of transpositions.


For example, for permutation
$\begin{pmatrix}
1 & 2 & 3 & 4\\ 
2 & 3 & 1 & 4
\end{pmatrix}$
decomposition is
$\begin{pmatrix}
1 & 3
\end{pmatrix}
\begin{pmatrix}
1 & 2
\end{pmatrix}$
but why are $\begin{pmatrix}
1 & 3
\end{pmatrix}$ and $\begin{pmatrix}
1 & 2
\end{pmatrix}$ transpositions? Aren't they not even cycles?
 [/INST] A cycle of length 2 is called a transposition, of course it switches (transposes) two elements.
$$
(1 3) =
\left(
\begin{matrix}
1 & 2 & 3 & 4 \\
3 & 2 & 1 & 4
\end{matrix}
\right)
$$
</s>