The commutator subgroup of Dn: Is it generated by ρ2?

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In summary, the author has worked out the commutators of Dn for n=3,4,5 and 6, but cannot prove it. They speculate that the group may require 2 generators. They also mention that < and > can be used to write the generators.
  • #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.
 
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  • #2
Hey TDA120! :smile:

Shall we try to find the commutators of Dn?
Let's start with [ρ,σ], where ρ is a rotation over an angle α=2π/n and σ is a reflection in the x-axis.
Do you know how to find that?
Perhaps with the use of complex numbered functions?
Did you perhaps already do an exercise where you combined a rotation with a reflection, and/or a reflection with another reflection?

Once you have that, we can try to generalize to [ρklσ]...
And any other possible commutators (which are those?).

Btw, did you know that ρkσ=σρ-k?
That might make it easier to calculate the commutator.
 
  • #3
Thanks! I think this meant I could make some new steps.!?
ρkρlσρ-klσ)-1=
ρkρlσρ-kσρ-l=
ρk+lσσρk-l=
ρ2k

So, every commutator will be either id or ρ2k
As 2k is a multiple of ρ2, [Dn,Dn] is generated by ρ2.
So [Dn,Dn] = [itex]\left\langle[/itex] ρ2k [itex]\right\rangle[/itex] = {id, ρ2, ρ4, …, ρn-2} if n is even and

[Dn,Dn] = [itex]\left\langle[/itex] ρ2k [itex]\right\rangle[/itex] = Dn+ if n is odd.

Then the order of [Dn,Dn] is 2n/n = 2 if n is odd.
And the order of [Dn,Dn] is n/2 if n is even.
Dn/[Dn,Dn] = 2n/(n/2) = 4

Each group of order 4 is isomorphic with the Cyclic group C4 or V4.
 
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  • #4
Yep! :approve:

Btw, are you allowed to use that ρkσ=σρ-k?
Or do you need to prove that?

And did you consider other commutators?
Such as a reflection with a reflection?

Are you clear on which of C4 or V4 you're isomorphic with?
 
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  • #5
Hi I like Serena!,

Thanks again!

Yes I three-double checked with all sorts of commutators.

Is this enough proof for ρkσ = σρ-k
ρkσ *ρkσ = id as ρkσ is a reflection
Then ρkσ *ρk = σ
And ρkσ = σρ-k

If n is even, Dn /[Dn,Dn] consists of four elements, id and three others: ρ, σ and ρσ as Dn has been divided by all even powers of ρ. This group has two generators, just as V4.

Does this hold?
 
  • #6
Sounds good! ;)

I only think it's not trivial that the group requires 2 generators and not just 1.
How do you know that it requires at least 2 generators?

Oh and those generator angles should be written as just \langle and \rangle, so you get: ##\langle \rho^{2k} \rangle##.
You can also use < and >, although I usually feel they look less nice, getting: ##< \rho^{2k} >##.
 
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  • #7
[itex]\langle[/itex]Right. I wasn’t thinking the right way; thanks![itex]\rangle[/itex]
 
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  • #8
:smile:
 

FAQ: The commutator subgroup of Dn: Is it generated by ρ2?

1. What is an isomorphism?

An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures. It essentially means that two structures have the same underlying mathematical properties and can be transformed into each other without losing any information.

2. What does it mean for a group to be abelian?

A group is considered abelian if its group operation is commutative, meaning that the order in which the elements are multiplied does not change the result. In other words, for all elements a and b in the group, a * b = b * a.

3. What is the significance of group Dn in abstract algebra?

Group Dn, also known as the dihedral group, is a group of symmetries that is commonly used in abstract algebra to study the properties of regular polygons. It is a finite group with 2n elements, and is generated by two elements: a rotation and a reflection.

4. How do you determine if a group Dn is isomorphic to an abelian group?

To determine if a group Dn is isomorphic to an abelian group, you need to look at the properties of the group. If the group operation is commutative, then it is abelian. Additionally, if the group has n elements, it must also have n distinct elements of order 2 in order to be isomorphic to an abelian group.

5. What are some real-world applications of isomorphisms in abstract algebra?

Isomorphisms are used in various fields of science and technology, such as cryptography, chemistry, and physics. In cryptography, isomorphisms are used to encrypt and decrypt messages, while in chemistry, they are used to study the symmetries of molecules. In physics, isomorphisms are used to study symmetries in quantum mechanics and other physical systems.

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