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Let $G$ be a finite group and let $V$ be an irreducible complex representation of $G$. Does there exist an element $g \in G$ which acts on $V$ with distinct eigenvalues? If true, can you provide a proof/reference, and if false, a counterexample?

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If $g\in G$ has order $n$, then its eigenvalues are all $n$th roots of unity. So if $V$ is a representation with degree greater than $n$, the eigenvalues of $g$ on $V$ can't all be distinct.

There are plenty of examples of groups with irreducible representations whose degree is greater than the largest order of any element of the group, and so these will certainly give counterexamples. For example, the symmetric group $S_6$ has irreducible representations of degree 9,10 and 16.

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