character grouppg 29 which uses as the target instead of just . This is useful while studying complex tori because the character group of the lattice in a complex torus is canonically isomorphic to the dual torus via the Appell-Humbert theorem. That is,We can express explicit elements in the character group as follows: recall that elements in can be expressed asfor . If we consider the lattice as a subgroup of the underlying real vector space of , then a homomorphismcan be factored as a mapThis follows from elementary properties of homomorphisms. Note thatgiving us the desired factorization. As the groupwe have the isomorphism of the character group, as a group, with the group of homomorphisms of to . Since for any abelian group , we haveafter composing with the complex exponential, we find thatwhich is the expected result. Examples Finitely generated abelian groups Since every finitely generated abelian group is isomorphic tothe character group can be easily computed in all finitely generated cases. From universal properties, and the isomorphism between finite products and coproducts, we have the character groups of is isomorphic tofor the first case, this is isomorphic to , the second is computed by looking at the maps which send the generator to the various powers of the -th roots of unity . Orthogonality of characters Consider the matrix A = A(G) whose matrix elements are where is the kth element of G. The sum of the entries in the jth row of A is given by if , and . The sum of the entries in the kth column of A is given by if , and . Let denote the conjugate transpose of A. Then . This implies the desired orthogonality relationship for the characters: i.e., , where is the Kronecker delta and is the complex conjugate of . See also Pontryagin duality References See chapter 6 of Number theory Group theory Representation theory of groups