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Stabiliser Groups of a vertex/edge of a square

  1. Sep 27, 2012 #1
    1. The problem statement, all variables and given/known data

    Given the dihedral group of symmetries of a square; what is the stabiliser group of a vertex (or edge)?

    2. Relevant equations
    The stabiliser group is G_x={g[itex]\in[/itex]G|gx=x}
    I guess for a vertex/edge that means the transformations in D4 (generated by reflection in x axis and rotation by 90°)

    3. The attempt at a solution
    Doodling in my notebook, I have determined that the only way that the order of vertices/edges can be changed is by reversing them - which means that the only unique elements of G that map a vertex/edge to itself is the identity, and one other - which depends on the vertex/edge, but is basically a rotation combined with a reflection, that reverses the order of the vertices and then rotates it back to the relevant spot... are theere any others? and how can I do it algebraically to demonstrate a complete stabilising subgroup?
     
  2. jcsd
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