# How to prove that the symmetry group of a regular polygon has only 1 and 2 dim irredu

1. Nov 20, 2009

### wdlang

how to prove that the symmetry group of a regular polygon has only 1 and 2 dim irreducible representations?

2. Nov 24, 2009

### morphism

Re: how to prove that the symmetry group of a regular polygon has only 1 and 2 dim ir

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!

Last edited: Nov 24, 2009
3. Nov 25, 2009

### wdlang

Re: how to prove that the symmetry group of a regular polygon has only 1 and 2 dim ir

Thanks a lot

maybe you mean the rotation subgroup is of index 2?

i will think of it.

4. Nov 27, 2009

### morphism

Re: how to prove that the symmetry group of a regular polygon has only 1 and 2 dim ir

Yup - sorry! (The reflection subgroup has order 2!)

5. Oct 25, 2010

### wdlang

Re: how to prove that the symmetry group of a regular polygon has only 1 and 2 dim ir

but i still can not figure out the proof

any hint?

is there any theorem?