Editing Character group (section)

== Examples ==

=== Finitely generated abelian groups ===
Since every [[finitely generated abelian group]] is isomorphic to<blockquote><math>G \cong \mathbb{Z}^n \oplus \bigoplus_{i=1}^m \mathbb{Z}/a_i</math></blockquote>the character group can be easily computed in all finitely generated cases. From universal properties, and the isomorphism between finite products and coproducts, we have the character groups of <math>G</math> is isomorphic to<blockquote><math>\text{Hom}(\mathbb{Z},\mathbb{C}^*)^{\oplus n}\oplus\bigoplus_{i=1}^k\text{Hom}(\mathbb{Z}/n_i,\mathbb{C}^*)</math></blockquote>for the first case, this is isomorphic to <math>(\mathbb{C}^*)^{\oplus n}</math>, the second is computed by looking at the maps which send the generator <math>1 \in \mathbb{Z}/n_i</math> to the various powers of the <math>n_i</math>-th roots of unity <math>\zeta_{n_i} = \exp(2\pi i/n_i)</math>.