**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

be a homomorphism to an abelian group A.

Then there exists a homomorphism

so that f can be created as a composition

of

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

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.