Dihedral Group/Group Presentation

Theorem
The dihedral group $D_n$ is generated by two elements $\alpha$ and $\beta$ such that:
 * $(1): \quad \alpha^n = e$
 * $(2): \quad \beta^2 = e$
 * $(3): \quad \beta \alpha = \alpha^{n - 1} \beta$

That is:
 * $D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$

Proof
Let $\alpha$ be rotation by $\dfrac {2 \pi} n$.

Let $\beta$ be reflection across the $y$ axis.