Proof regarding a subgroup of a cyclic group.

Question

I am practicing Abstract Algebra using course resources from when it was taught at my university last year. I am confused, in part (b) as to how $x^a=e$. Could someone explain this?

Answer 1

Perhaps a change in notation might help.

Here $$H=\{\xi\in G\mid \xi^a=e\}.$$

Let $x\in H$. Then, by definition, $x\in G$ such that $x^a=e$, since $x$ has to be one such $\xi\in H$.