Irreducible representations of the Dn group

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Summary:

Is is true that the dihedral group Dn does not have an irreducible representation with a dimension higher than two?

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Is is true that the dihedral group ##D_n## does not have an irreducible representation with a dimension higher than two?
 

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martinbn
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You can see that in following way. The group has a normal Abelian subgroup of index 2. If you have an irreducible representation of the dihedral group say ##V##, restrict it to the subgroup, then it is a sum of one dimensional representation ##V=\oplus V_i##. The action of the full group is determined by the action of a representative of the nontrivial coset (remember index two). Pick one of the one dimensional spaces say ##V_1##, the representative either sends it to itself or to another one dimensional subsapce say ##V_2##. Then in the first case ##V_1## is invariant under the full group, in the second case ##V_1\oplus V_2## is. So it must be that ##V## is either one or two dimensional.
 
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