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Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism.

Any hints how to even start?

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  • $\begingroup$ I suppose what you are looking for is proposition 25 section 8.2 in serre's representation theory finite groups. $\endgroup$ – random123 Feb 14 '16 at 18:44
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    $\begingroup$ Look at how an irrep restricts to $C_n$. $\endgroup$ – Qiaochu Yuan Feb 14 '16 at 20:06
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    $\begingroup$ It follows from the facts that there is abelian subgroup of index $2$ and all irreducible representations of abelian groups have dimension $1$. $\endgroup$ – Derek Holt May 26 '16 at 10:14
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Each irreducible representation of $D_n$ is spanned by at most $[G:H]$ independent vectors for every abelian subgroup $H<D_n$

Let $H=<r> \subset D_n$.

$H$ is clearly abelian and $[G:H]=2$, hence the result.

if $|H|=n$ is even, there are 4 irreducible representations of dimension 1 and $n/2 - 1$ irreducible representations of dimension 2.

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