Proof on relationship between generators and order of a cyclic group.

Question

Why do we have $a^{nq} = e$ in this proof?

Answer 1

For any element $\;a\;$ of a group of order $\;n\;$ we have by Lagrange's theorem that

$$a^n=e\implies a^{nq}=\left(a^n\right)^q=e^q=e$$

Answer 2

"Let $G$ be a cyclic group of order $n$ (...)"

You assume that $G$ is a cyclic group of order $n$, so $a^{nq}=\left (a^n\right )^q=e^q=e$ by definition.