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 ''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 , 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 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 (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 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 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