I Irreducible representations of the Dn group

[Robin04]

TL;DR Summary

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

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

 

[fresh_42]

Seems so:
https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dihedral_groups

 

[Robin04]

Seems so:
https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dihedral_groups

Thank you!

 

[martinbn]

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.