1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:
    fyg7b4.jpg

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Dihedral Group on a square
  1. Dihedral group (Replies: 1)

  2. Dihedral groups (Replies: 1)

Loading...