How do i write a permutation as a product of disjoint cycles ?
I know that in order to determine a cycle we need to start with the smallest element and move on till the mapping points to itself.Then start with the next non repeating smallest element..But how to write this as a product of disjoint cycles?
Best Answer
First, we note that writing it as a product of disjoint cycles means that each number appears only once throughout all of the cycles.
We see that $1\mapsto 5$, $5\mapsto 3$, $3\mapsto 2$, $2\mapsto 1$. So, we can express this in cycle notation as $$(1532).$$ Now, we see what is left over... well, that is just $4$, which is fixed by the permutation in question. So, the permutation can be written as
$$(1532)(4),\mbox{ or equivalently, just } (1532).$$