Prove Symmetry Group of Regular Polygon Has 1 & 2 Dim Irreducible Reps

wdlang
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how to prove that the symmetry group of a regular polygon has only 1 and 2 dim irreducible representations?
 
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The group in question, i.e. the dihedral group, has an abelian subgroup of index 2 (the one generated by the reflection). Thus any irreducible representation is at most 2 dimensional. I'll let you fill in the details. Post back if you need more help!
 
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morphism said:
The group in question, i.e. the dihedral group, has an abelian subgroup of index 2 (the one generated by the reflection). Thus any irreducible representation is at most 2 dimensional. I'll let you fill in the details. Post back if you need more help!

Thanks a lot

maybe you mean the rotation subgroup is of index 2?

i will think of it.
 


wdlang said:
maybe you mean the rotation subgroup is of index 2?
Yup - sorry! (The reflection subgroup has order 2!)
 


morphism said:
The group in question, i.e. the dihedral group, has an abelian subgroup of index 2 (the one generated by the reflection). Thus any irreducible representation is at most 2 dimensional. I'll let you fill in the details. Post back if you need more help!

but i still can not figure out the proof

any hint?

is there any theorem?
 

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