What does it mean to find elements in $S_9$ that are "not cycles"?

Question

I came across this wording in the following question. Some clarification on what this means and how to approach this problem would be helpful. Thanks!

Answer 1

The notation $|\mu|$ means the order of $\mu$ in $S_9$.

Hint:

• Every permutation is a product of disjoint cycles

• The order of a product of disjoint cycles is the lcm of their orders

Solution:

$\mu = (1234)(56)$ has order $4$ but is not a cycle. Other examples are $(1234)(5678)$ and $(1234)(56)(78)$. As for $\omega$, an element of order $5$ must be a product of disjoint cycles of order $5$, but in $S_9$ there are no two disjoint cycles of order $5$ because that would required at least $10$ objects to permute.