Finding presentations of (finite) metacyclic groups: Doubts about one of D. L. Johnson's books.

Question

I am reading the book "Presentations of Groups", by D.L Johnson and I have doubts about page 88, Proposition 1.

Why is there a need to prove that $N \cong \mathbb{Z}_m$?

Because it's given that $N=\langle x\rangle$ and it's also given that $x^m=e$.

Please refer to the book.

Thanks!