Product of Generating Elements of Dihedral Group

Theorem
Let $D_n$ be the dihedral group of order $2 n$.

Let $D_n$ be defined by its group presentation:
 * $D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$

Then for all $k \in \Z_{\ge 0}$:
 * $\beta \alpha^k = \alpha^{n - k} \beta$

Proof
The proof proceeds by induction.

For all $k \in \Z_{\ge 0}$, let $\map P k$ be the proposition:
 * $\beta \alpha^k = \alpha^{n - k} \beta$

$\map P 0$ is the case:

Thus $\map P 0$ is seen to hold.

Basis for the Induction
We have:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P r$ is true, where $r \ge 1$, then it logically follows that $\map P {r + 1}$ is true.

So this is the induction hypothesis:
 * $\beta \alpha^r = \alpha^{n - r} \beta$

from which it is to be shown that:
 * $\beta \alpha^{r + 1} = \alpha^{n - r - 1} \beta$

Induction Step
This is the induction step:

So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall k \in \Z_{\ge 0}: \beta \alpha^k = \alpha^{n - k} \beta$