# Analytic Number Theory/Characters and Dirichlet characters

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## Definitions, basic properties

Definition 4.1

Let ${\displaystyle G}$ be a finite group. A character of G is a function ${\displaystyle f:G\to \mathbb {C} }$ such that

1. ${\displaystyle \forall \sigma ,\tau \in G:f(\sigma \tau )=f(\sigma )f(\tau )}$ and
2. ${\displaystyle \exists \rho \in G:f(\rho )\neq 0}$.

Lemma 4.2:

Let ${\displaystyle G}$ be a finite group and let ${\displaystyle f:G\to \mathbb {C} }$ be a character. Then

${\displaystyle \forall \sigma \in G:|f(\sigma )|=1}$.

In particular, ${\displaystyle \forall \sigma \in G:f(\sigma )\neq 0}$.

Proof:

Since ${\displaystyle G}$ is finite, each ${\displaystyle \sigma \in G}$ has finite order ${\displaystyle n:=\mathrm {ord} (\sigma )}$. Furthermore, let ${\displaystyle \rho \in G}$ such that ${\displaystyle f(\rho )\neq 0}$; then ${\displaystyle f(\rho )=f(\sigma )f(\sigma ^{-1}\rho )}$ and thus ${\displaystyle f(\sigma )\neq 0}$. Hence, we are allowed to cancel and

${\displaystyle |f(\sigma )|=|f(\sigma ^{n+1})|=|f(\sigma )|^{n+1}\Rightarrow |f(\sigma )|=1}$.${\displaystyle \Box }$

Lemma 4.3:

Let ${\displaystyle G}$ be a finite group and let ${\displaystyle f,g:G\to \mathbb {C} }$ be characters. Then the function ${\displaystyle h:G\to \mathbb {C} ,h(\tau ):=f(\tau )\cdot g(\tau )}$ is also a character.

Proof:

${\displaystyle h(\sigma \tau )=f(\sigma \tau )g(\sigma \tau )=f(\sigma )g(\sigma )f(\tau )g(\tau )=h(\sigma )h(\tau )\neq 0}$,

since ${\displaystyle \mathbb {C} }$ is a field and thus free of zero divisors.${\displaystyle \Box }$

Lemma 4.4:

Let ${\displaystyle G}$ be a finite group and let ${\displaystyle f:G\to \mathbb {C} }$ be a character. Then the function ${\displaystyle g:G\to \mathbb {C} ,g(\tau ):={\frac {1}{f(\tau )}}}$ is also a character.

Proof: Trivial, since ${\displaystyle \forall \tau \in G:f(\tau )\neq 0}$ as shown by the previous lemma.${\displaystyle \Box }$

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 ${\displaystyle G}$ be a finite group. Then the group

${\displaystyle \{f:G\to \mathbb {C} |f{\text{ is a character }}\}=\mathrm {Hom} (G,\mathbb {C} ^{\times })}$

is called the character group of ${\displaystyle G}$.

## Required algebra

We need the following result from group theory:

Lemma 4.6

Let ${\displaystyle G}$ be a finite Abelian group, let ${\displaystyle H\leq G}$ be a subgroup of order ${\displaystyle n\in \mathbb {N} }$, and let ${\displaystyle \tau \in G\setminus H}$ such that ${\displaystyle k\in \mathbb {N} }$ is the smallest number such that ${\displaystyle \tau ^{k}\in H}$. Then the group

${\displaystyle N:=\{\tau ^{n}\sigma |\sigma \in H\}}$

is a subgroup of ${\displaystyle G}$ containing ${\displaystyle H}$ of order ${\displaystyle k\cdot n}$.

Proof:

Since ${\displaystyle G}$ is the disjoint union of the cosets of ${\displaystyle H}$, ${\displaystyle N}$ is the disjoint union ${\displaystyle \bigcup _{j=0}^{n-1}\tau ^{j}H}$, as ${\displaystyle \rho H=H\Leftrightarrow \rho \in H}$ and ${\displaystyle \tau ^{l}H=\tau ^{m}H\Leftrightarrow \tau ^{l-m}\in H\Leftrightarrow k|(l-m)}$. Hence, the cardinality of ${\displaystyle N}$ equals ${\displaystyle k\cdot n}$.

Furthermore, if ${\displaystyle \tau ^{l}\sigma ,\tau ^{m}\rho \in N}$, then ${\displaystyle \tau ^{l}\sigma (\tau ^{m}\rho )^{-1}=\tau ^{l-m}\sigma \rho ^{-1}\in N}$, and hence ${\displaystyle N}$ is a subgroup.${\displaystyle \Box }$