Cycle Decomposition of $S_{3}$ Suppose the cycle decomposition $\sigma$ has the following values on ${1,2,3}$,
$\sigma(1)=3$,
$\sigma(2)=2$,
$\sigma(3)=1$.
This cycle decomposition is expressed as $(1 3)$.
Was the third value of $\sigma$ needed to know this cycle decomposition ?
In other words, in order to form a cycle decomposition from any permutation, is it necessary to know the value of that permutation at each and every element of the set being permuted ?

A permutation on $n$ elements, is uniquely determined by is value on $n-1$ of those elements since the there will only be one choice for the value of the $n$-th element.
