 Solution

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.