Cyclic & Dihedral Groups

C_n rotates, D_n adds mirrors — the symmetry groups behind orbit counting.

The idea

The cyclic group $C_n$ and the dihedral group $D_n$ are the symmetry groups of the regular polygons, and they are the standard first examples of finite groups.

Consider a regular $n$-gon and the motions that leave it occupying its original outline. Rotation by $1/n$ of a full turn is one; call it $r$. Repeating it gives the rotations $e, r, r^{2}, \dots, r^{n-1}$, with $r^{n} = e$. These $n$ rotations form the cyclic group $C_n$. Reflecting the polygon across one of its $n$ mirror lines is also a symmetry, and no rotation equals a reflection. The $n$ rotations together with the $n$ reflections form the dihedral group $D_n$, of order $2n$.

Composition behaves differently in the two groups: composing two rotations adds their angles, while composing a rotation with a reflection follows no such rule, and the result depends on which of the two is applied first.

In any group, the order of an element $g$ is the smallest integer $m > 0$ with $g^{m} = e$.

Ways to work on it

• Walkthrough. Orders of C_n and D_n, abelian-ness, and the order of r^k.
• Practice. Compute the order of r^k in C_n via n/ (n,k).
• Hardest. Count the elements of a given order in a dihedral group.

Not sure where to start? Take the ten-question placement test.