C/Cx/Exponential mapping/Isomorphism theorem/Example

The complex exponential function

$(\mathbb {C} ,0,+)\longrightarrow (\mathbb {C} ^{\times },1,\cdot ),z\longmapsto \exp z,$

is a surjective group homomorphism. The kernel is ${}\mathbb {Z} 2\pi {\mathrm {i} }$. Due to the isomorphism theorem, there exists a canonical isomorphism

${}\mathbb {C} /\mathbb {Z} 2\pi {\mathrm {i} }\cong \mathbb {C} ^{\times }\,.$