Transform a permutation from cycle notation to standard form

Question

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

the object can viewed as:

$\sigma =$

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

Now given the following permutation in cycle notation, I want to convert it to standard form as above.

consider the cycle: $(1,2)(3,5)(4,1,3)$ Now if it didn't have the repeats of $(1,3)$ I would be able to write it out as above

Answer 1

Since you go from right to left,

$(1,2)(3,5)(4,1,3)$ maps $1\mapsto3\mapsto5$, $2\mapsto1$, $3\mapsto4$, $4\mapsto1\mapsto2$, and $5\mapsto3$,

so it's $\pmatrix {1&2&3&4&5\\5&1&4&2&3}.$