How Many Orbits Are Formed by Dihedral Group Actions on Colored Squares?

[Ted123]

Homework Statement

Let G=D_4 (the group of symmetries (reflections/rotations) of a square) and let X=\{ \text{colourings of the edges of a square using the colours red or blue} \} so a typical element of X is:

What is the size of X?

Let G act on X in the obvious way. You are given that G has 6 orbits on X. Find a representative for each G-orbit, and its size.

The Attempt at a Solution

Obviously there are going to be various ways of colouring the edges of a square, but how can I be sure that I have them all or is there a quicker way to find the size?

How do I find a representative for each G-orbit?

 

[Joffan]

Size of X: Each side has a choice of two colours, and there are 4 sides.

Orbits of G - Some properties of X will not be affected by the symmetry operations. This will naturally lead to separate orbits.

 

[Ted123]

Size of X: Each side has a choice of two colours, and there are 4 sides.

Orbits of G - Some properties of X will not be affected by the symmetry operations. This will naturally lead to separate orbits.

The definition of orbit is \text{orb}_G(x)= \{ gx : g\in G \}
Saying that G has 6 orbits on X: does this mean there are 6 x's?

So by saying 'find a representative' does it mean find 6 different x's?

 

[Joffan]

It means (as I read it) there are six categories of x that will be mapped only onto their own category by the symmetries in G.

For example: The x that consists of all blue edges will be mapped to itself by any G. It is an orbit of size one.

 

[Ted123]

It means (as I read it) there are six categories of x that will be mapped only onto their own category by the symmetries in G.

For example: The x that consists of all blue edges will be mapped to itself by any G. It is an orbit of size one.

Is the size of X 16?

 

[Joffan]

That's what I get. It's a conveniently small-enough number that you can actually draw them all out, too, if you need to.