Let $G$ be the multiplicative group of complex numbers of modulus $1$ and let $G_n$ (with $n$ a positive integer) be the subgroup consisting of the $n$-th roots of unity. For positive integers $m$ and $n$, I want to show that $G/G_m$ and $G/G_n$ are isomorphic groups.
I can't even prove that the natural map is well defined. Please help me. Thanks.
For every $n$ the map $\Bbb{C}^{\times}\ \longrightarrow\ \Bbb{C}^{\times}:\ z\ \longmapsto\ z^n$ is a surjective group homomorphism with kernel $G_n$, so by the first isomorphism theorem $G/G_n\cong G$.