In
mathematics, a character is (most commonly) a special kind of
function from a
group to a
field (such as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s). There are at least two distinct, but overlapping meanings.
Other uses of the word "character" are almost always qualified.
Multiplicative character
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 , usually the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
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
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
); 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
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vect