Is the Klein Four Group Identical to the Dihedral Group D2?

[ChrisVer]

Can someone please read the attached file and tell me his ideas? I want to be sure I understand the action of the Klein Four group...Is my interpretation correct?

I have some problem though. The Klein Four group has 4 elements, and it is able to describe the reflective symmetries of a square (as I interpreted it).
The dihedral group however, $D_2$ for example should be the same group right? (if I see the one $Z_2$ of the KF group as a cyclic group and the other $Z_2$ are reflection). $D_2$ however does not describe a square...

 

[lpetrich]

That should be easy to work out from the definition of dihedral group D(n): generators a and b with
aⁿ = b² = e
a.b = b.a^-1

Note what happens when n = 2. What is a^-1 in that case?

 

[ChrisVer]

Hi,

How is it connected to the dihedral group?

Because the D(2) is of order 4, and since it's not cyclic C(4) it has to be (at least isomorphic to) the Klein Four group?

D_2 = (e , a, b, ab)
and a^2=e \Rightarrow a^{-1}=a
So I wonder for n=2, there is no distinction between rotations and reflections (rotations are done by 180^o)

 

[lpetrich]

I'll work out D2 or Dih(2). It has
a² = b² = e
a.b = b.a^-1
But from the definition of a, a^-1 = a, and thus, a.b = b.a. Thus, all the group's elements commute with each other.

The n = 2 rotation and reflection elements:

• Identity
• 180d rotation
• Two reflections

They all commute with each other.

Rotation: $$ \begin{pmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $$
Reflection: $$ \begin{pmatrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{pmatrix} $$
where θ is the rotation angle. For D2, it is 0 and 180d. In general, $\theta_{rot}(k) = 2\pi k / n$ and
$\theta_{refl}(k) = \theta_0 + 2\pi k / n$ for k = 0, 1, ..., n-1, where θ₀ is arbitrary.