# Character (mathematics)

In mathematics, a **character** is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings.^{[1]} Other uses of the word "character" are almost always qualified.

A **multiplicative character** (or **linear character**, or simply **character**) on a group *G* is a group homomorphism from *G* to the multiplicative group of a field (Artin 1966), usually the field of complex numbers. If *G* is any group, then the set Ch(*G*) of these morphisms forms an abelian group under pointwise multiplication.

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

In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "character theory" and one-dimensional characters are also called "linear characters" within this context.