Character group

In mathematics, a character group is the group of representations of a 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 arise in the related context of character theory. Whenever a group is represented by matrices, the function defined by the trace of the matrices is called a character; however, these traces do not in general form a group. Some important properties of these one-dimensional characters apply to characters in general:

The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform. For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.

Let G be an abelian group. A function ${\displaystyle f:G\rightarrow \mathbb {C} \backslash \{0\}}$ mapping the group to the non-zero complex numbers is called a character of G if it is a group homomorphism from ${\displaystyle G}$ to ${\displaystyle \mathbb {C} ^{\times }}$—that is, if ${\displaystyle \forall g_{1},g_{2}\in G\;\;f(g_{1}g_{2})=f(g_{1})f(g_{2})}$.

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