 
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 Nth root z of unity. Then,
for each g G and ^G, there is a unique integer 1 r N such that (g) = zr
. We
