dihedral group (thing)

The Dihedral group D_n is the symmetry group of the regular n-gon¹.

This group has 2n elements. They are the rotations given by the powers of r, rotation anti-clockwise through 2pi/n, and the n reflections given by reflection in the line through a vertex (or the midpoint of an edge) and the centre of the polygon.

         |
 \       |       /
   \ ----|---- /
   | \   |   / |
   |   \ | /   |
------------------
   |   / | \   |
   | /   |   \ |
   / ----|---- \
 /       |       \     

Let s denote one of these reflections.

It's not hard to see that

Dn={1,r,r2,...,rn-1,s,rs,...,rn-1s}

and that we have the relations: rⁿ=1 s²=1 and srs^-1=r^-1. In fact the Dihedral group is given by these generators and relations.

If we label the vertices of the polygon by {1,2,...,n} then each symmetry of the polygon gives rise to a permutation of {1,2...,n}. This gives a homomorphism to the Symmetric group D_n->S_n. It' s easy to see that it is an injection.

In the case n=4 this is what we get.

  
  2          1
    --------
   |        |
   |        |
----------------
   |        |
   |        |
    --------
  3          4           

Let s be reflection in the x-axis and let r be as before. Then s corresponds to the permutation (23)(14) and t corresponds to (1234). Thus D₄ is a Sylow 2-subgroup of S₄.

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1. There is some disagreement about notation here. Some people write D_2n for what I've called D_n