Zig-zag lemma

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 Template:Harv, 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 $\chi _{1},\chi _{2},\ldots ,\chi _{n}$ 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

$G:=\left\{\left.{\begin{pmatrix}a&b\\0&1\end{pmatrix}}\ \right|\ a>0,\ b\in \mathbf {R} \right\}.$

Functions f_u : G → C such that $f_{u}\left({\begin{pmatrix}a&b\\0&1\end{pmatrix}}\right)=a^{u},$ where u ranges over complex numbers C are multiplicative characters.

• Consider the multiplicative group of positive real numbers (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.