[[Schur's lemma]]
# Irreducible representations of abelian groups are 1-dimensional

Let $G$ be an [[Abelian group]].
As a corollary of [[Schur's lemma]], if $\Gamma:G\to\mathrm{GL}(V)$ is a (complex) [[Irrep]]
then it is a [[1-dimensional irrep]]. #m/thm/rep 
Moreover, any [[Abelian representation|abelian irrep]] is 1-dimensional.

> [!check]- Proof
> If $G$ is abelian, then so are all its irreps.
> Let $\Gamma$ be an abelian irrep of $G$ on $V$,
> so $\Gamma(h)\Gamma(g) = \Gamma(g)\Gamma(h)$ for every $g,h \in G$,
> and therefore $\Gamma(h)$ 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 $\Gamma$ to be an irrep.
> <span class="QED"/>

See also [[Abelian representation#Main theorem]].

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