Group Theory 1 - Dihedral Group of Order 2n

Let’s continue our journey on the Dihedral groups of order \(2n\).

First, let’s establish a few things:

A Note On Symmetry - Just Isometry

As we go through these groups, we’re playing the game, what motions of the polygon leave the polygon looking the same (what are its symmetries)?

Reconsider “symmetry” as a manipulation of the plane. Then you can rephrase the game of looking for symmetries as looking for isometries of the plane.

\(D_4\), the \(2\)-sided polygon

This group is the case when \(n=2\), then the total order of the group is \(4\). The operations are still identity, reflection, and rotation.

It is important to note that this \(2\)-gon has 2 vertices and 2 edges, meaning the two vertices do NOT share an edge, unlike the \(1\)-sided polygon. While I draw them as curved edges, this is only a representation of otherwise very close straight lines.

This group is isomorphic to the Klein 4-group, this also means that the group is isomorphic to the cyclic group of order 2 :)