Dihedral Groups

Definition of Dihedral Group

The dihedral group is the group of symmetries of a regular polygon. In abstract algebra, it's a classic example of a finite, nonabelian group. Dihedral groups are also a great example of how transformations can be thought of as elements in a group. Here, a transformation that involves picking up the polygon, and flipping or rotating it around, and then placing it back so that it lines up with the starting position. These transformations form a group. After all, since each transformation returns the group to its original position, so does the composition of any two transformations. It is a nonabelian group (i.e. non-commutative) because a flip followed by a rotation is different than a rotation followed by a flip.

Example of a Dihedral Group

This group is easy to work with computationally, and provides a great example of a connection between groups and geometry. For example, imagine an equilateral triangle. There are a total of 6 different transformations (try to imagine what they are.) And it's nonabelian because a flip-then-rotate is not the same as a rotate-then flip. It's good practice to make a list of all the transformations, and determine which one is the identity element, and check that every transformation has an inverse.

Consider this...

The dihedral groups give you groups that act on geometry. Is there any way to use our knowledge of regular polygons to help us study these groups?