decompose the permutation $\sigma$ in cycle notation into an object

Question

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

As an object a permutation $\sigma$ looks like this:

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

They can also be written in terms of cycle notation which looks like this for the above permutation:

$(1,2,3),(4), and (5,6)$

I want to take the following permutation in cycle form and convert it into its form as an object.

$(1,3)(1,2)(4,5)$

It is the duplication of the $1'$s in two separate orbits that are throwing me off. I have been taught to go from right to left

Answer 1

Just follow how the cycles act on elements. For instance (working right to left), we have for $(1\;2)(1\;3)\;(2\;3)$ $$ \begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix} $$ For instance, tracking the action of this product acts on 2. First, it sends 2 to 3 then it sends that 3 to 1 then it sends that 1 to 2. Therefore, it fixes 2. Follow the action of elements similarly. [Note again, you may have been taught to compose left to right, rather than my right to left. It depends on the convention you are learning.]

Answer 2

You could multiply $(13)(12)(45)=((13)((12))(45)=(123)(45)$. From there the object form is easy.