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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)?

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If "representation" means "finite-dimensional representation," then it turns out that you can find nonabelian $G$ whose only representations are trivial!

Groups that have faithful linear representations are called linear, and a theorem due to Malcev asserts that a finitely generated linear group is residually finite. Taking the contrapositive, it follows that a finitely generated group that is not residually finite cannot be linear. And inspecting the proof of Malcev's theorem, you can squeeze out another result: a finitely generated group which admits a nontrivial representation admits a nontrivial map to a finite group. So to find a finitely generated group with no nontrivial representations, it suffices to find a finitely generated group with no subgroups of finite index. The Higman group was the first known such group.

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 $\mathbb{R}$.

I'm not sure how to deal with infinite-dimensional irreducible representations.

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  • $\begingroup$ I did mean finite-dimensional representations (edited accordingly) $\endgroup$ – user3533 Apr 11 '13 at 10:11

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