**1. The problem statement, all variables and given/known data**
My challenge is as follows:

Let D

_{n} be the dihedral group (symmetries of the regular n-polygon) of order 2n and let ρ be a rotation of D

_{n} with order n.

(a) Proof that the commutator subgroup [D

_{n},D

_{n}] is generated by ρ

^{2}.

(b) Deduce that the abelian made D

_{n,ab} is isomorphic with {±1} in case n is odd, and with V

_{4} (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 $\displaystyle f: G \to A$ be a homomorphism to an abelian group A.

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

$\displaystyle G \overset{\pi}{\to} G_{ab} \overset{f_{ab}}{\to}A$

of $\displaystyle \pi: G \to G_{ab}$ with f

_{ab}.

Here G

_{ab} is the group that is made abelian from G.

**Corollary**: Every homomorphism f: S

_{n}->A to an abelian group A is the composition of $\displaystyle S_n \to \{\pm 1\} \overset{h}{\to} A$ of the sign function with a homomorphism h: {±1} -> A

**3. The attempt at a solution**
I have worked out [D

_{n}, D

_{n}] for n=3,4,5 and 6 and have noticed the above described pattern. I just cannot proof it.