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Let $G$ be a finite group and $\mathbb{k}$ is algebraically closed with characteristic zero. Let $H$ be an Abelian subgroup of $G$. Show that the degree of any irreducible representation $V$ of $G$ is bounded by the index of $H$ in $G$.

Given any irreducible representation $V$ of $G$, I know I can get a representation of $H$ by restricting: $V|_H$. But $H$ is Abelian, so $V|_H$ will be a direct sum of $1$-dimensional, and hence irreducible, representations of $H$...

I just don't see how am I going to be able to say something meaningful about the degree of a representation of $G$. Clearly it will have something to do with counting cosets, so it'd be nice if $H$ were normal, but that need not be the case (I'm guessing, given the way the problem is written.) Any help is appreciated

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    Thank you, @GeoffRobinson. If you'd like to submit this as an answer I will gladly accept it2012-10-16

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