Let $G$ be a finite group and let $Z$ be its centre. Show that if $\rho$ is a complex irreducible representation of $G$, assumed to be faithful (that is, the kernel of $\rho$ is trivial), then $Z$ is cyclic.

Now assume that $G$ is a p-group (that is, the order of $G$ is a power of the prime $p)$, and assume that $Z$ is cyclic. If $\rho$ is a faithful representation of $G$, show that some irreducible component of $\rho$ is faithful.

[You may use without proof the fact that, since $G$ is a p-group, $Z$ is non-trivial and any non-trivial normal subgroup of $G$ intersects $Z$ non-trivially.]

Deduce that a finite $p$-group has a faithful irreducible representation if and only if its centre is cyclic.