# Nonabelian group with all irreducible representations one-dimensional

All irreducible representations of an abelian group are one-dimensional. For a finite group, the coverse is also true - if all irreducible representations are one-dimensional then the group is abelian.

Is there a nonabelian group $G$ such that all of its finite-dimensional complex irreducible representations are one-dimensional (such group $G$ is necessarily infinite of course)?

If "representation" means "finite-dimensional representation," then it turns out that you can find nonabelian $G$ whose only representations are trivial!
Note also that if $G$ is residually finite then whether or not $G$ is finitely generated it's still true that $G$ is abelian iff its finite-dimensional irreducible representations are all $1$-dimensional (exercise). Many familiar groups are residually finite, so this rules out examples that are too easy.
Another strategy for constructing examples is to find infinite simple groups with cardinality strictly larger than that of $\mathbb{R}$ (the same as the cardinality of $\text{GL}_n(\mathbb{C})$), since any nontrivial finite-dimensional representation of such a group is necessarily faithful. Examples include $\text{PSL}_3(F)$ where $F$ is a field of cardinality strictly larger than $\mathbb{R}$.