Solution
= Solution
If $g$ belongs to the <centre of a group>, then $\rho(g)$ commutes with $\rho(h)$ for every $h\in G$. The <Schur lemma> says that an endomorphism of a finite-dimensional complex <irreducible representation> commuting with the representation is scalar. Therefore
$$
\boxed{\rho(g)=\lambda_gI\quad(g\in Z).}
$$
Since $g$ has finite order, $\lambda_g$ is a <root of unity> and has modulus one.