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Let $V$ be a representation of a finite group $G$ with $V$ being finite dimensional. Fix a $g \in G$. Is it necessarily true that $\dim V \geq \operatorname{ord} g$?

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    If $G$ is abelian then all its irreducible representations have dimension 1, but (unless it's the one-element group) it has elements of order greater than 1.2012-02-14

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No, not at all. Let $V$ be a one-dimensional vector space, and let $G$ be the two-element cyclic group, where the generator for $G$ acts as multiplication by $-1$. Then $V$ is one-dimensional, but $G$ has an element of order two.