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When I turn on my monitor, the brand name fills the screen. But since I mounted my monitor upside down so I can watch it in bed looking up, the power-on screen is upside down.

But I noticed that it could also be the result of being flipped both vertically and horizontally in 2 mirrors.

Both group actions result in the same result. Is there some kind of higher level group that they belong to?

Or is the fact that both functions have the same output of no real interest, like the fact that 4 + 3 equals 7 and 5 + 2 also equals 7?

Shaun
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Miss Understands
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4 Answers4

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You have two group elements $a,b$ such that $a^2=e$ and $b^2=e$ (the flips). They combine to make a third element $ab=c$ such that $c^2=e$ (the rotation). This gives the presentation

$$\langle a,b,c\mid a^2, b^2, c^2, ab=c\rangle.$$

This is a presentation of the Klein four group.

Shaun
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Generally speaking, an orthogonal transformation of the Cartesian plane consists of those linear transformations representing rotations centered on the origin and reflections across lines through the origin.

The group consisting of all orthogonal transformation is an infinite group called the orthogonal group (in dimension 2) and it is denoted $O(2)$.

The Klein 4-group, which you describe, is a subgroup of $O(2)$, consisting of the linear transformations defined by the following four matrices $$\underbrace{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}_{\text{the identity}} \qquad \underbrace{\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}}_{\text{reflection across $y=0$}} \qquad \underbrace{\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}}_{\text{rotation by $180^\circ$}} \qquad \underbrace{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}}_{\text{reflection across $x=0$}} \qquad $$

Lee Mosher
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This is the Klein four group - the symmetries of a rectangle.

Ethan Bolker
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You should note that the rotations (around the origin) form a group:

  • if you do two rotations in a row, you obtain a rotation
  • also the identity (not moving anything) can be viewed as a rotation of angle 0°

But the reflections (over an axis passing through the origin) do not form a group:

  • if you do two reflections in a row, you obtain a rotation, not a reflection
  • the identity can not be seen as a reflection over an axis

If you combine the rotations and reflexions, you do get a group. It is called the orthogonal group $\mathrm{O}(2)$, the group of linear tranformations preserving distances (note that all linear tranformations must preserve the origin).

The group of rotations is a subgroup of the orthogonal group called the special orthogonal group $\mathrm{SO}(2)$, the group of linear transformations preserving distances and orientation. It is actually "half" the group, as the orientation has two possible states, preserved or flipped. (FYI, the orientation of an orthogonal transformation is linked to the determinant. Also, the "$2$" stands for $2$ dimensions).

As the other answers said, if you consider only the rotation by 0° or 180° and the reflexions over the X-axis or the Y-axis, you have a subgroup of $\mathrm{O}(2)$ of order $4$ (size $4$) and this subgroup corresponds to the Klein 4-group.

For me, the interesting fact that you are looking for is the role of $\mathrm{SO}(2)$ in $\mathrm{O}(2)$, being a subgroup of index $2$ ("half" the group) and more concretely the surprising fact that the combination of $2$ reflections always gives a rotation.

Vincent
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