Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 19F / a

Choose a nonzero vector in an irreducible representation. Its finite group orbit spans a finite-dimensional invariant subspace, which must be the whole representation. The representing matrices of an abelian group commute. Choose an eigenspace of one matrix; all other matrices preserve it. Repeatedly choosing an eigenspace for the restrictions of the remaining matrices gives a common eigenvector. Its line is invariant under the whole group. Irreducibility therefore makes the representation that line, proving every irreducible complex representation of a finite abelian group is one-dimensional.

A faithful representation of dimension one embeds $A$ into ${\mathbb{C}}^{\times }$. Every image element is a $|A|$th root of unity by Lagrange's theorem, so the image is a subgroup of the cyclic group of those roots. It is cyclic, and faithfulness makes $A$ cyclic. Conversely a generator of a cyclic group of order $m$ can be sent to $e^{2\pi i/m}$, giving an injective one-dimensional character. Thus such a faithful representation exists exactly when $A$ is a cyclic group.