More exam preparation.
Let $A$ be a normal subgroup of a finite group $G$ and $V$ an irreducible representation of $G$. Show that either $\text{Res}_A^G V$ is isotypic (a sum of copies of one irreducible representation of $A$) or that $V$ is induced from some proper subgroup of $G$.
Now, normally when I am asked to prove $P \vee Q$, I see a reasonable path to proving either $\neg P \Rightarrow Q$ or $\neg Q \Rightarrow P$, but I don't see how to do either here. It seems hard to make use of $\neg Q$ as a hypothesis, and I don't see how to relate $\neg P$ as a hypothesis to $Q$ (mainly because I don't see how to produce the needed subgroup $H$). The material most likely to be relevant to this question is, I guess, Mackey restriction or Mackey irreducibility, but I don't quite see how either of these can be applied.