Section 12 Dihedral Groups

For \(n \ge 3\), the dihedral group \(\mathsf{D}_n\) is the set of rigid symmetries of a regular \(n\)-sided polygon (an “\(n\)-gon”). For consistency, we will arrange our \(n\)-gon as points around the unit circle, with 1 at position (1,0) and the points ordered counter clockwise, as seen here.

Let \(r\) be the rotation by \(2 \pi/n\) counter-clockwise, and let \(f\) be the reflection (flip) over the \(x\)-axis. Then \(\mathsf{D}_n\) has the following \(2n\) elements \[ \mathsf{D}_n = \{1, r, r^2, \ldots, r^{n-1}, f, rf, r^2f, \ldots, r^{n-1}f\}. \] It is generated by \(r\) and \(f\) subject to the relations \[ r^n = 1, \qquad f^2 = 1, \qquad f r = r^{-1} f = r^{n-1} f. \]

The group is non-cyclic and nonabelian. The set of rotations \[ R_n = \langle r \rangle = \{1, r, r^2, \ldots, r^{n-1}\} \] is a cyclic subgroup of order \(n\). Since it is half the group it is a normal subgroup of index \([\mathsf{D}_n:R_n] = 2\).

The dihedral group \(\mathsf{D}_n\) can be represented as a subgroup of the symmetric group \(\mathsf{S}_n\) by identifying each the generators with the following permutations (in disjoint cycle notation): \[ r = (1, 2, 3, \ldots, n), \qquad f = (2,n)(3,n-1)(4,n-2) \cdots. \] When \(n = 3\), \(\mathsf{S}_3 \cong \mathsf{D}_3\). Otherwise, for \(n > 3\), \(\mathsf{D}_n\) is a proper subgroup of \(\mathsf{S}_n\).