TITLE: Irreducible representations and Abelian subgroups
QUESTION [2 upvotes]: There is a theorem in representation theory which is surprising to me: the dimension of irreducible (complex) representation of finite group is not greater that $(G:H)$ - an index of Abelian subgroup $H\subset G$. Can anyone provide a reference or a proof of this statement?

REPLY [0 votes]: Note that if $V$ is any irreducible representation of $G$ and $H$ is a subgroup of $G$, then $V$ appears in the induced representation $\mathrm{Ind}(\mathrm{Res}(V))$, and this in fact implies that every irreducible representation of $G$ appears (as a summand of) $\mathrm{Ind}(W)$ for some irreducible representation of $H$.
If $H$ is abelian, every irreducible representation is one dimensional, and $\mathrm{Ind}(W)$ has dimension $[G:H]$, so the claim follows.
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