# Character of a group

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A homomorphism of the given group into some standard Abelian group . Usually, is taken to be either the multiplicative group of a field or the subgroup

of . The concept of a character of a group was originally introduced for finite groups with (in this case every character takes values in ).

The study of characters of groups reduces to the case of Abelian groups, since there is a natural isomorphism between the groups and , where is the commutator subgroup of . The characters form a linearly independent system in the space of all -valued functions on . A character extends uniquely to a character of the group algebra . The characters are one-dimensional linear representations of over ; the concept of a character of a representation of a group coincides in the one-dimensional case with the concept of a character of a group. Sometimes characters of a group are understood to mean characters of any of its finite-dimensional representations (and even to mean the representations themselves).

A character of a topological group is a continuous homomorphism . If is a locally compact Abelian group, then its characters separate points, that is, for any , , there exists a character such that . For Hausdorff Abelian groups this assertion is not true, in general (see [3]). A character of an algebraic group over an algebraically closed field is a rational homomorphism .

In number theory an important role is played by the characters of the multiplicative group of the residue ring modulo , which correspond one-to-one to Dirichlet characters modulo : To a character there corresponds the Dirichlet character given by the formula