Permutation order

Question

How do you put non disjoint permutation cycles into disjoint cycle form? For Example the permutation in non disjoint cycle form (1352)(34)? How do you form disjoint cycle for from this?

Answer 1

Hint: $(13)(34) = (31)(34) = (341) = (134) = (14)(13)$.

Although in this case since the permutation is small you can just make a table $x\to\sigma(x)$ and follow the orbits.

Answer 2

You just need to figure out the image of each number. For example start with $1$: in this case $1\to 3$. Now write down $(1 \ 3$. Next start with $3$, so you can stay in the same cycle: here $3\to4$. So, now we have $(1 \ 3 \ 4$. Keep going until you reach $1$ again, then close parentheses. If you exhausted all the numbers in that cycle you'll have finished. If you haven't, open a new cycle starting with some element that didn't appear in the previous one. Keep going until you exhaust all elements this way. Note that disjoint cycles commute, so it doesn't matter what order you write them in once you've found them all.