Editing Character group (section)

In [[mathematics]], a '''character group''' is the [[group (mathematics)|group]] of [[group representation|representation]]s of an [[abelian group]] by [[complex number|complex]]-valued [[function (mathematics)|function]]s. These functions can be thought of as one-dimensional [[matrix (mathematics)|matrix]] representations and so are special cases of the group [[character (mathematics)|character]]s that arise in the related context of [[character theory]]. Whenever a group is represented by matrices, the function defined by the [[trace (linear algebra)|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:
* Characters are invariant on [[conjugacy class]]es.
* The characters of irreducible representations are orthogonal.
The primary importance of the character group for [[finite group|finite]] abelian groups is in [[number theory]], where it is used to construct [[Dirichlet character]]s.  The character group of the [[cyclic group]] also appears in the theory of the [[discrete Fourier transform]]. For [[locally compact abelian group]]s, the character group (with an assumption of continuity) is central to [[Fourier analysis]].