## Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself

## Problem 221

Let $p$ be a prime number. Let

\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.

Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.

Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.