Irreducible Representations of Abelian Group

Theorem
Let $\left({G, \cdot}\right)$ be a finite abelian group.

Then $\rho: G \to \operatorname{GL} \left({V}\right)$ is an irreducible linear representation iff $\dim \left({V}\right) = 1$.

Proof
One implication holds for every type of group and is proved in Representation of Degree One is Irreducible.

Take $g\in G$, since $G$ is an abelian group, $\rho(g)\rho(h)=\rho(h)\rho(g)$ for all $h\in G$.

Then from Conmutative Linear Transformation is G-Module Homomorphism and Endomorphisms of G-Module follows that $\rho(g)=\lambda_g \operatorname{Id}_V$. Hence $\rho(g)$ is the linear mapping of multipliying by $\lambda_g$.

Thus $\rho(g)(v)=\lambda_g v$; and any vector subspace of $V$ of dimension $1$ is invariant. Thus only the representations of degree $1$ can be irreducible.