(a) Let $A$ be a normal subgroup of a finite group $G$, and let $V$ be an irreducible representation of $G$. Show that either $V$ restricted to $A$ is isotypic (a sum of copies of one irreducible representation of $A)$, or else $V$ is induced from an irreducible representation of some proper subgroup of $G$.

(b) Using (a), show that every (complex) irreducible representation of a $p$-group is induced from a 1-dimensional representation of some subgroup.

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