Consider a finite group $G$ of order $n$ and $\mathbb{C}G$ its regular module.
Show that
(1) No irreducible representation of $\mathbb{C}G$ has dimension greater than $n/k$ where $k$ is the order of any element in $G$.
(2) Every irreducible representation of $\mathbb{C}D_{2n}$ has dimension 1 or 2.
Also,
(3) Fix an element $g$ of order $k$, consider the map $l_g:\mathbb{C}G \rightarrow \mathbb{C}G$ by $l_{g}(h)=gh$. Show that the eigenvalues of $l_g$ are precisely the k-th roots of unity, each with multiplicity $n/k$.
For (1), I don't know where to begin. But for (2), I tried $D_8$ and the result indeed shows that the dimensions are either 1 or 2. But I don't know how to prove the general result.
For (3), I tried to let $gh=\lambda h$ for $\lambda \in \mathbb{C}$. Then $g^{k}h^k=\lambda^{k}h^{k}$ which gives $(\lambda^{k}-1)h^{k}=0$ since $g$ has order $k$. I guess(?) this shows $\lambda$ is the $k$-th roots of unity. How can I show the multiplicity?
You can prove (1) using Frobenius reciprocity. Let $\chi$ be an irreducible character of $G$ and $H = \langle g \rangle$ a cyclic subgroup, and let $\rho$ be an irreducible constituent of $\chi_H$. Then $H$ is abelian, so $\rho$ has degree $1$, and by Frobenius reciprocity $\chi$ is a constituent of $\rho^G$, which has degree $n/k$, where $k=|g|$.
This argument works for any abelian subgroup of $G$, which gives a stronger result.