How to use Cayley's theorem to prove the following?

Question

How to prove the underlined part in this example using the Cayley's Theorem? Thanks for help. I have no idea on it.

Answer 1

The underlined part follows directly from Cauchy's Theorem (not Cayley's theorem, which is about embedding into a symmetric group):

Proving Cauchy's theorem (group theory)

For $p=5$ and $p\mid | G|$, Cauchy's Theorem gives an element of order $p=5$ in $G$.

Answer 2

As @DietrichBurde noted in their answer and a comment below, it's really Cauchy's theorem that matters here. Every element must have an order dividing $5$, i.e. equal to $1$ or $5$. But they can't all be of order $1$.