Product of Generating Elements of Quaternion Group

Theorem
Let $Q = \Dic 2$ be the quaternion group:
 * $\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$

Then for all $k \in \Z_{\ge 0}$:
 * $b a^k = a^{-k} b$

Proof
The proof proceeds by induction.

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

$\map P 0$ 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:
 * $b a^r = a^{-r} b$

from which it is to be shown that:
 * $b a^{r + 1} = a^{-r - 1} b$

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:
 * $\forall k \in \Z_{\ge 0}: b a^k = a^{-k} b$