Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 19I / iii / Solution

If $\rho$ is a faithful representation, $g\mapsto {\lambda }_g$ embeds $Z$ into the multiplicative group of complex numbers. Its image is a finite group of roots of unity, hence lies in the cyclic group of $n$th roots of unity and is itself cyclic. Thus $Z$ is cyclic.