Irreducible representations of abelian groups are 1-dimensional

Let $𝐺$ be an Abelian group. As a corollary of Schur's lemma, if $Γ : 𝐺 \to G L (𝑉)$ is a (complex) Irrep then it is a 1-dimensional irrep. #m/thm/rep Moreover, any abelian irrep is 1-dimensional.

Proof

If $𝐺$ is abelian, then so are all its irreps. Let $Γ$ be an abelian irrep of $𝐺$ on $𝑉$ , so $Γ (ℎ) Γ (𝑔) = Γ (𝑔) Γ (ℎ)$ for every $𝑔, ℎ \in 𝐺$ , and therefore $Γ (ℎ)$ is a multiple of the identity for all $ℎ \in 𝐺$ , so every subspace of $𝑉$ is invariant under $𝐺$ . Thus $𝑉$ must be 1-dimensional in order for $Γ$ to be an irrep.