In mathematics, 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 {{Harv|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 (characters whose 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 <math>\chi_1, \chi_2, \ldots, \chi_n</math> are different characters on a group ''G'' then from <math>a_1\chi_1 + a_2\chi_2 + \cdots + a_n\chi_n = 0</math> it follows that <math>a_1 = a_2 = \cdots = a_n = 0.</math>
==Examples==
*Consider the (''ax'' + ''b'')-group :: <math> G := \left\{ \left. \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\ \right|\ a > 0,\ b \in \mathbf{R} \right\}.</math> : Functions ''f''<sub>''u''</sub> : ''G'' → '''C''' such that <math>f_u \left(\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\right)=a^u,</math> where ''u'' ranges over complex numbers '''C''' are multiplicative characters.
* Consider the multiplicative group of positive real numbers ('''R'''<sup>+</sup>,·). Then functions ''f''<sub>''u''</sub> : ('''R'''<sup>+</sup>,·) → '''C''' such that ''f''<sub>''u''</sub>(''a'') = ''a''<sup>''u''</sup>, where ''a'' is an element of ('''R'''<sup>+</sup>, ·) and ''u'' ranges over complex numbers '''C''', are multiplicative characters.
==References== * {{citation|title=Galois Theory|series=Notre Dame Mathematical Lectures, number 2|authorlink=Emil Artin|first=Emil|last= Artin|year=1966|publisher = Arthur Norton Milgram (Reprinted Dover Publications, 1997)|isbn=978-0-486-62342-9}} Lectures Delivered at the University of Notre Dame
Category:Group theory
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