solution verification write the permutation in cycle notation and find the inverse

Question

Definition 1.4.1: A permutation of a set is a bijection (one-to-one and onto) : → .

Write the permutation in cycle form

$\sigma =$

$$ \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 3 & 4 & 6 & 5 \\ \end{matrix} $$

My guess is that I leave out $3,4$ and obtain:

$(1,2),(5,6)$

The inverse is just:

$$ \begin{matrix} 2 & 1 & 3 & 4 & 6 & 5 \\ 1 & 2 & 3 & 4 & 5 & 6 \\ \end{matrix} $$

Answer 1

If you are right, and you are, it is its own inverse , because the transpositions are disjoint, hence commute. And transpositions have order $2$.