Cyclic Group

A cyclic Group is any group that can be generated by a single element.

$|G|\ge \omega \Rightarrow G\cong (\mathbb{N},+)$

Take $x$ st $⟨x⟩=G$
Have $x^n=x^m\Rightarrow 1=x^{m-n}⟹m=n$ (else |G| divides (m-n)).
Thus all distinct
Hence the result.

$|G|=m\in \mathbb{N}⟹G\cong {\mathbb{Z}}_m$