Prove that the set $\{f\in \mathfrak G|f(p)=w\}$(where $w$ is a $r$th root of unity) is a coset of the subgroup $\{f\in \mathfrak G|f(p)=1\}$.

Question

Consider the group of units of integer modulo $k$ that is $\Bbb Z_k^*$. There consider the character as a map $f:\Bbb N \to \Bbb C$ s.t $f(n) = \begin{cases} f(\bar n) & \text{if $(n,k)=1$ } \\ 0 & \text{otherwise} \end{cases}$

So the character is a group homomorphism on $\Bbb Z_k^*$.

Now consider the character group $\mathfrak G$. My question is:

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Let $\bar p$ has order $r$ in $\Bbb Z_k^*$. Then the values of $f(p)$ are the $r$th root of unity.

Prove that the set $\{f\in \mathfrak G|f(p)=w\}$(where $w$ is a $r$th root of unity) is a coset of the subgroup $\{f\in \mathfrak G|f(p)=1\}$.

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I was studying John Binder's notes to prove Dirichlet's Theorem.

There it is written at page 14 Analytic Number Theory and Dirichlet's Theorem

Answer 1

In general, if $f:G\to H$ is a group homomorphism, and $b\in\text{Im}\, f$, then $f^{-1}(b)=\{a\in G:f(a)=b\}$ is a coset (left or right) of $K=\text{Ker}\, f$. This is because, if $a_0\in f^{-1}(b)$ then $a\in f^{-1}(b)$ iff $f(a)=f(a_0)$ iff $aa_0^{-1}\in K$ iff $a\in Ka_0 $.