Cyclic-center obstruction to a faithful irreducible representation
= Cyclic-center obstruction to a faithful irreducible representation
If a finite group has a faithful irreducible complex representation, every central element acts as a scalar by <Schur lemma>. Faithfulness embeds the centre into $\mathbb C^\times$, and every finite subgroup of $\mathbb C^\times$ is cyclic. Hence the group centre must be cyclic.