mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a multiplicative character (or linear character, or simply character) on a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''G'' is a

group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...

from ''G'' to the

multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...

of a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, usually the field of

complex numbers 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 form a ...

. If ''G'' is any group, then the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 commu ...

under pointwise multiplication. This group is referred to as the

character group 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 a ...

of ''G''. Sometimes only ''unitary'' characters are considered (characters whose

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

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 character In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi: \mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: # \chi(ab) = \ch ...

s 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 exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...

, i.e. if

\chi_1, \chi_2, \ldots, \chi_n

are different characters on a group ''G'' then from

a_1\chi_1 + a_2\chi_2 + \cdots + a_n\chi_n = 0

it follows that

a_1 = a_2 = \cdots = a_n = 0.

Examples

*Consider the (''ax'' + ''b'')-group ::

G := \left\.

: Functions ''f''_''u'' : ''G'' → C such that

f_u \left(\begin
a & b \\
0 & 1  \end\right)=a^u,

where ''u'' ranges over complex numbers C are multiplicative characters. * Consider the multiplicative group of positive

real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...

(R⁺,·). Then functions ''f''_''u'' : (R⁺,·) → C such that ''f''_''u''(''a'') = ''a''^''u'', where ''a'' is an element of (R⁺, ·) and ''u'' ranges over complex numbers C, are multiplicative characters.

References

* Lectures Delivered at the University of Notre Dame Group theory {{group-theory-stub