Finding the maximum possible order for an element in $S_5$

Question

I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order.

But my question is, do I have to write all elements of $S_5$, write them as disjoint cycles, and then find the largest least common multiple, or is there a shortcut?

$S_5$ has $5!$ elements and I would not like writing all of these permutations out...

I have read the other answers on here but I have not seen anything to help me with this question.

For example, here (https://math.stackexchange.com/a/231893/133156) the answer lists six disjoint cycles of $S_5$, how did he get there without writing them all out?

Answer 1

You only need to write out all possible structures for disjoint cycles. They correspond to the (additive) partitions for $5$:

$5 = 5 $ corresponds to one $5$-cycle.

$5 = 4 + 1 $ corresponds to one $4$-cycle and one $1$-cycle, but $1$-cycles can be ignored.

$5= 3 + 2 $ corresponds to one $3$-cycle and one $2$-cycle.

etc...