A p-group is a group of order $p^d$ where p is a prime.
If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible representation of the center which would map a generator of the center to $e^{2\pi i/m}$. Could we then induce a representation on the whole group? If so, how do we know this is faithful and irreducible? If not, how else could we prove this?
The induced representation is faithful, but not necessarily irreducible. But, since the centre of the $p$-group $P$ is cyclic, the unique subgroup $K$ of $Z(P)$ with $|K|=p$ is contained in every nontrivial normal subgroup of $P$. Since the induced representation is faithful, at least one of its irreducible constituents does not have $K$ in its kernel, and then that consituent must be faithful.