Let $N$ be a proper normal subgroup of a finite group $G$ and let $U$ be an irreducible complex representation of $G$. Show that either $U$ restricted to $N$ is a sum of copies of a single irreducible representation of $N$, or else $U$ is induced from an irreducible representation of some proper subgroup of $G$.

Recall that a $p$-group is a group whose order is a power of the prime number $p$. Deduce, by induction on the order of the group, or otherwise, that every irreducible complex representation of a $p$-group is induced from a 1-dimensional representation of some subgroup.

[You may assume that a non-abelian $p$-group $G$ has an abelian normal subgroup which is not contained in the centre of $G$.]