Character of an abelian group

Remarks

Example: for $G = \Z/n\Z$ and an integer $j$, the map $\chi_j(k) = e^{2\pi i jk/n}$ is well defined, because $e^{2\pi i j(k + n)/n} = e^{2\pi i jk/n}$. It is a character, since $\chi_j(k + l) = e^{2\pi i j(k+l)/n} = \chi_j(k)\chi_j(l)$. Non-example: the map $\Z \to \C$, $k \mapsto 2^k$, satisfies the homomorphism rule, but its values do not have absolute value $1$, so it is not a character. The characters of $G$ form an abelian group under pointwise multiplication $(\chi\psi)(x) = \chi(x)\psi(x)$, with identity the trivial character $x \mapsto 1$ and inverse $\overline{\chi}$. This group is the character group, often written $\widehat{G}$. For a finite group every homomorphism to $\C^\times$ takes values in $\mathbb{T}$, so some texts define characters as homomorphisms to $\C^\times$.