Question:
Let $p$ be a prime number. Let $G_n=C_{p^n}$ be the cyclic group of order $p^n$ with generator $x_n$. We define $\varphi:G_n \rightarrow G_{n+1}$ by $\varphi(x_n^a)=x_{n+1}^{pa}$. Using the above construction we obtain a group $G$, called a quasicyclic group and usually denoted $C_{p^\infty}$. Prove that $C_{p^\infty} \cong H$ where $H=\{ z \in \mathbb{C}^\times: \exists n \: z^{p^n}=1 \}$.
The lecturer didn't include the "above construction" and I can't find any more detail on the net. Anyone know how to construct this group from the homomorphisms given?