Character (group theory)

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. 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.

Multiplicative characters are linearly independent, i.e. if ${\displaystyle \chi _{1},\chi _{2},\ldots ,\chi _{n}}$ are different characters on a group G then from ${\displaystyle a_{1}\chi _{1}+a_{2}\chi _{2}+\ldots +a_{n}\chi _{n}=0}$ it follows that ${\displaystyle a_{1}=a_{2}=\cdots =a_{n}=0}$.

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