 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. ThereforeSince $g$ has finite order, ${\lambda }_g$ is a root of unity and has modulus one.