# Analytic Number Theory/Characters and Dirichlet characters

## Definitions, basic properties

[edit | edit source]**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.

**Definition 4.5**

Let be a finite group. Then the group

is called the **character group** of .

## Required algebra

[edit | edit source]We need the following result from group theory:

**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.