- #1

TDA120

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1. Homework Statement

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. Homework 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.

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. Homework 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.

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