Analytic Number Theory/Characters and Dirichlet characters

Definitions, basic properties 0 100% #FFFFFF Definition 4.1 Let be a finite group. A character of G is a function such that and . Lemma 4.2: Let be a finite group and let be a character. Then . In particular, . Proof: Since is finite, each has finite order . Furthermore, let such that ; then and thus . Hence, we are allowed to cancel and . Lemma 4.3: Let be a finite group and let be characters. Then the function is also a character. Proof: , since is a field and thus free of zero divisors. Lemma 4.4: Let be a finite group and let be a character. Then the function is also a character. Proof: Trivial, since as shown by the previous lemma. The previous three lemmas (or only the first, together with a few lemmas from elementary group theory) justify the following definition. 0 100% #FFFFFF Definition 4.5 Let be a finite group. Then the group is called the character group of . Required algebra We need the following result from group theory: 0 100% #FFFFFF Lemma 4.6 Let be a finite Abelian group, let be a subgroup of order , and let such that is the smallest number such that . Then the group is a subgroup of containing of order . Proof: Since is the disjoint union of the cosets of , is the disjoint union , as and . Hence, the cardinality of equals . Furthermore, if , then , and hence is a subgroup. Theorems about characters Dirichlet characters