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Homework Help: Dihedral Group on a square

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data

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

    What is the size of [itex]X[/itex]?

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

    3. 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?
  2. jcsd
  3. Jan 30, 2012 #2
    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.
  4. Jan 30, 2012 #3
    The definition of orbit is [tex]\text{orb}_G(x)= \{ gx : g\in G \}[/tex]
    Saying that G has 6 orbits on X: does this mean there are 6 [itex]x[/itex]'s?

    So by saying 'find a representative' does it mean find 6 different [itex]x[/itex]'s?
  5. Jan 30, 2012 #4
    It means (as I read it) there are six categories of [itex]x[/itex] that will be mapped only onto their own category by the symmetries in G.

    For example: The [itex]x[/itex] that consists of all blue edges will be mapped to itself by any G. It is an orbit of size one.
  6. Jan 30, 2012 #5
    Is the size of X 16?
  7. Jan 30, 2012 #6
    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.
    Last edited: Jan 30, 2012
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