Klein Four Group's action

[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.

 

[blue_raver22]

I can confirm that your interpretation of the Klein Four group's action is correct. The Klein Four group is a finite group that consists of four elements, each representing a different type of symmetry. These symmetries can be visualized as reflections of a square, as you have correctly interpreted.

However, it is important to note that while the Klein Four group and the dihedral group $D_2$ both have four elements, they are not the same group. The dihedral group is a more general group that describes the symmetries of a regular polygon, not just a square. In fact, the dihedral group $D_4$ would be a more appropriate group to describe the symmetries of a square.

It is also worth mentioning that while the Klein Four group and the dihedral group $D_2$ both have two cyclic subgroups, they are not the same subgroups. The two cyclic subgroups of the Klein Four group are isomorphic to the group $\mathbb{Z}_2$, while the two cyclic subgroups of the dihedral group $D_2$ are isomorphic to the group $\mathbb{Z}_4$. This difference in subgroups reflects the difference in the underlying symmetries of the two groups.

In summary, your interpretation of the Klein Four group's action is correct, but it is important to distinguish between the Klein Four group and the dihedral group $D_2$ as they are not the same group and have different applications.