There are several meanings of the word character in mathematics, although all are related to the idea of using complex numbers to study a more abstract algebraic structure.

Number theoretic characters

If G is group, a character is a group homomorphism into the multiplicative group of complex numbers. If A is an abelian group, then the set Ch(A) of these morphisms forms a group under the operation

χ_aχ_b=χ_ab.

This group is referred to as the character group. Sometimes only unitary characters are considered (so that the image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen a special case of this definition.

Representation characters

If f is a representation of a group G, then the character of the representation is the function from G to the complex numbers given by the trace of f. In general, the trace is neither a group homomorphism, nor does the set of traces form a group. The study of representations by means of their characters is called character theory.

Algebraic characters

If A is an abelian algebra over the complex numbers, a character of A is an algebra homomorphism into the complex numbers. If in addition, A is a *-algebra, then a character is a *-homomorphism into the complex numbers.