Generators of multiplicative group

Question

Let G be the multiplicative group generated by the complex number e^$(2πiθ)$, θ a real number. For what values of θ is G a finite group? What is its order in that case?

How would one proceed to solve this question?

Answer 1

By definition, $G$ is cyclic with generator $e^{2\pi i \theta}$. Hence, the order of $G$ is equal to the order of the element $e^{2\pi i \theta}$, i.e. the smallest positive integer $n$ such that $e^{2\pi i \theta n} =1$ (in case such $n$ does not exist, the order is not finite). Now what do you know about the exponential function?