1. The problem statement, all variables and given/known data(adsbygoogle = window.adsbygoogle || []).push({});

My challenge is as follows:

Let Dn be the dihedral group (symmetries of the regular n-polygon) of order 2n and let ρ be a rotation of Dn with order n.

(a) Proof that the commutator subgroup [Dn,Dn] is generated by ρ2.

(b) Deduce that the abelian made Dn,ab is isomorphic with {±1} in case n is odd, and with V4 (the Klein four-group) in case n is even.

2. Relevant equations

The Fundamental theorem on homomorphisms

Fundamental theorem on homomorphisms - Wikipedia, the free encyclopedia

Proposition: Let [itex]f:[/itex] G [itex]\rightarrow[/itex] [itex]A[/itex] be a homomorphism to an abelian group A.

Then there exists a homomorphism [itex]f_{ab}: G_{ab}=G/[G,G] \to A[/itex] so that f can be created as a composition

[itex]G \overset{\pi}{\to} G_{ab} \overset{f_{ab}}{\to}A[/itex]

of [itex]\pi: G \to G_{ab}[/itex] with fab.

Corollary: Every homomorphism f: Sn->A to an abelian group A is the composition of [itex]S_n \to \{\pm 1\} \overset{h}{\to} A[/itex] of the sign function with a homomorphism h: {±1} -> A

3. The attempt at a solution

I have worked out [Dn, Dn] for n=3,4,5 and 6 and have noticed the above described pattern. I just cannot proof it.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Isom. of group Dn abelian

**Physics Forums | Science Articles, Homework Help, Discussion**