I have the following question:
Let $m>1$ be a positive integer and $G:=G(m):={(\mathbb{Z}/m\mathbb{Z})}^{*}$. Let $p\in\mathbb{P}$ with the property $p\nmid m$. Let $\widehat{G}:=\text{Char}(G)$ be the group of all characters modulo $m$. Let $\overline{p}$ be the image of $p$ in $G$ and let $f(p)$ denote the order of $\overline{p}$ in $G$.
Why is $H:=\{\chi\in \widehat{G}\ \big|\ \chi(p)=w;\ w\in\mathbb{C}\ \text{a fixed}\ f(p)\text{-th}\ \text{root of unity}\}$ a coset of the subgroup $U:=\{\chi\in \widehat{G}\ \big|\ \chi(p)=1\}\leq \widehat{G}$ ?
Thanks for the help!
Take two character $\chi_1$ and $\chi_2$ are in $\widehat{G}$. Take $H:=H_w$ where $w$ is a fixed $f(p)$ root of unity.
First assume that $\chi_1$ and $\chi_2$ are both in $H$ then the character $\chi_1\chi_2^{-1}$ verifies :
$$\chi_1\chi_2^{-1}(p)=\chi_1(p)\chi_2(p)^{-1}=ww^{-1}=1$$
So $\chi_1\chi_2^{-1}\in U$. By definition of coset of $\widehat{G}$ mod $U$ this shows that $H\subseteq [\chi_1]=\chi_1U$. Now for the converse, take $\chi\in U$ then :
$$[\chi_1\chi](p)=\chi_1(p)\chi(p)=w\times 1=w$$
That is $H=\chi_1 U$. This shows that all such $H_w$ are cosets for $\widehat{G}$ mod $U$.
To be complete we have to show that each coset $\chi_0U$ for $\widehat{G}$ mod $U$ is of this form but if we set $w_0:=\chi_0(p)$ we see that $w_0$ must be a $f(p)$-root of unity then by what we have done, we know that :
$$\chi_0U=H_{w_0} $$
So it follows that :
$$\widehat{G}/U=\{H_w|w\in\mathbb{C}\text{ with }w^{f(p)}=1\} $$