Rank of a linear transformation. Let G be a finite abelian group. Let N be the exponent of G and n be the order of G. Summary: Rank of a linear transformation. Let G be a finite abelian group. Let N be the exponent of G and n be the order of G. A function f : G C - {0} is called a character of G if it is a group homomorphism. If f is a character of a finite group, then each function value is a root of unity since all elements of a finite group have finite order. Notice that if g G, for any character of G we have (g)N = (gN ) = (e) = 1, so the values of lie among the Nth roots of unity. Characters on finite abelian groups were first studied in number theory, since number theory is a source of many interesting finite abelian gropus. A finite abelian group of order n has exactly n distinct characters which are denoted by f1, f2, ..., fn. f1 is the trivial representation, that is, f1(g) = 1 for all g G. It is called the principal character of G; the others are called nonprincipal characters, and fi(g) = 1 for some g G. The set of characters of G form an abelian group under multiplication called the character group. Let ^G denote the character group of G. Fix a primitive N-th root z of unity. Then, for each g G and ^G, there is a unique integer 1 r N such that (g) = zr . We Collections: Mathematics