Schur's lemma

See @baez1994gauge page 171.
One can use a basic result called Schur's Lemma. This states that if we have an irreducible group representation

any linear operator

that commutes with all the operators $p(g)$ must be a scalar multiple of the identity operator.

Consequence: if $G$ is abelian, then every $\rho (g)$ is a multiple of the identity, and every 1-dimensional subspace is invariant. Then the only irreducible representations of an abelian group must be 1-dimensional.