The problem is this:
Let $G$ be a finite group, $Q$ a normal subgroup of prime index.
(i) Let $\psi = \chi_V$ be the character of the $Q$ representation $V$. Show that either $\psi$ can be extended to $G$, or the induced representation is irreducible.
(ii) Show that if $Q$ is abelian, then every irreducible representation of $G$ is either one-dimensional or $p$-dimensional
For the first part, I've done one direction: If $V = \text{Res}(U)$, then $\text{Ind}(V)$ cannot be irreducible in this case - using Frobenius reciprocity and the Frobenius character formula. However, I'm stuck on the reverse direction.
For the second part, I have a strategy I'm a little bit skeptical of. Basically, I'm thinking you take an irrep $V$ of $G$, and restrict to $Q$. If this is an irreducible $Q$ rep, then we're done. Else, it has an irreducible $Q$ subrep $U$. I'm thinking that if I can show that $q \in Q$, $q \not = e$, then $gU \neq U$. In this case, I'm thinking I can then exhibit ${\rm Ind}(U)$ as a $G$ subrep of $V$, and then I would be done since it must be the whole of $V$. But, I'm pretty skeptical this totally works since I neither used the previous part, not the hypothesis that $Q$ is normal of prime index.
If someone could give me some insight into this question, I would appreciate it. I would also appreciate any general advice on dealing with the relationship between representations of some group $G$, and the representations of subgroups and quotients of $G$.