Rotations and reflections in dihedral group $D_8?$

Question

My question is:

In the dihedral group $D_8$, generated by $\alpha,\beta $, with $\alpha^8=\beta^2=e$, prove that $\alpha^7\beta=\beta\alpha$. Hence write $(\alpha^3\beta)(\alpha^2\beta)$ in the form $\alpha^p\beta^q$.

I can visualise the rotations $\alpha$ and reflections $\beta$ of the octagon $D_8$ but I am not sure how to prove the statement $\alpha^7\beta=\beta \alpha$. For the second part, through visualisation, I got $\alpha^2\beta^0$ but I am not sure how I would show this mathematically.

Thanks for any help!

Answer 1

For the second part, just walk $\beta$ through the various instances of $\alpha$ one by one:

$$\begin{align} \beta\alpha^2&=(\beta \alpha)\alpha \\ &= (\alpha^7\beta)\alpha \\ &= \alpha^7(\beta\alpha)\\ &=\alpha^{14}\beta\\ &=\alpha^6\beta. \end{align}$$

Answer 2

It is easy to visualise $\beta\alpha\beta^{-1}=\alpha^{-1}$ and therefore $\beta\alpha=\alpha^{-1}\beta=\alpha^{7}\beta$ for the first part.

Also, since $\beta\alpha\beta^{-1}=\alpha^{-1}$ we have $\beta\alpha^2\beta^{-1}=\alpha^{-2}$ and therefore $$(\alpha^3\beta)(\alpha^2\beta)=\alpha^3(\beta\alpha^2\beta)=\alpha^3\alpha^{-2}=\alpha.$$