In mathematics, a character group is the group of representations of an abelian group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in the related context of character theory. Whenever a group is represented by matrices, the function defined by the trace of the matrices is called a character; however, these traces do not in general form a group. Some important properties of these one-dimensional characters apply to characters in general:

The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform. For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.

Let ${\displaystyle G}$ be an abelian group. A function ${\displaystyle f:G\to \mathbb {C} ^{\times }}$ mapping ${\displaystyle G}$ to the group of non-zero complex numbers ${\displaystyle \mathbb {C} ^{\times }=\mathbb {C} \setminus \{0\}}$ is called a character of ${\displaystyle G}$ if it is a group homomorphism—that is, if ${\displaystyle f(g_{1}g_{2})=f(g_{1})f(g_{2})}$ for all ${\displaystyle g_{1},g_{2}\in G}$.

If ${\displaystyle f}$ is a character of a finite group (or more generally a torsion group) ${\displaystyle G}$, then each function value ${\displaystyle f(g)}$ is a root of unity, since for each ${\displaystyle g\in G}$ there exists ${\displaystyle k\in \mathbb {N} }$ such that ${\displaystyle g^{k}=e}$, and hence ${\displaystyle f(g)^{k}=f(g^{k})=f(e)=1}$.

Each character f is a constant on conjugacy classes of G, that is, f(hgh⁻¹) = f(g). For this reason, a character is sometimes called a class function.

A finite abelian group of order n has exactly n distinct characters. These are denoted by f₁, ..., f_n. The function f₁ is the trivial representation, which is given by ${\displaystyle f_{1}(g)=1}$ for all ${\displaystyle g\in G}$. It is called the principal character of G; the others are called the non-principal characters.

If G is an abelian group, then the set of characters f_k forms an abelian group under pointwise multiplication. That is, the product of characters ${\displaystyle f_{j}}$ and ${\displaystyle f_{k}}$ is defined by ${\displaystyle (f_{j}f_{k})(g)=f_{j}(g)f_{k}(g)}$ for all ${\displaystyle g\in G}$. This group is the character group of G and is sometimes denoted as ${\displaystyle {\hat {G}}}$. The identity element of ${\displaystyle {\hat {G}}}$ is the principal character f₁, and the inverse of a character f_k is its reciprocal 1/f_k. If ${\displaystyle G}$ is finite of order n, then ${\displaystyle {\hat {G}}}$ is also of order n. In this case, since ${\displaystyle |f_{k}(g)|=1}$ for all ${\displaystyle g\in G}$, the inverse of a character is equal to the complex conjugate.

There is another definition of character group^{pg 29} which uses ${\displaystyle U(1)=\{z\in \mathbb {C} ^{*}:|z|=1\}}$ as the target instead of just ${\displaystyle \mathbb {C} ^{*}}$. This is useful when studying complex tori because the character group of the lattice in a complex torus ${\displaystyle V/\Lambda }$ is canonically isomorphic to the dual torus via the Appell–Humbert theorem. That is,