Why do we say "Every permutation on $n$ symbols can be written as a product of disjoint cycles". I see this statement written like that almost everywhere and I think it's slightly incorrect.
For example, let $\sigma \in S_3$ and $\sigma = \left( \begin{matrix} 1 &2 & 3\\ 3 & 1 & 2 \end{matrix}\right)$ then how do you write $\sigma$ as product of disjoint cycles when $\sigma$ is simply a cycle $(1 \ 3 \ 2)$ itself?
I could understand if $\sigma \in S_4$ then you could write it as product of disjoint cycles as $\sigma = (1 \ 3 \ 2 )(4)$ but when it is in $S_3$, there is no product of disjoint cycles.
Am I on the right track? Or is there a way to write a cycle as product of disjoint cycles?
I think it should be something like "Every permutation on $n$ symbols is either a cycle or a product of disjoint cycles", is this correct?