Find an alternate generating set for Dih(2n)

Recall that $D_{2n} = \langle r,s \ |\ r^n = s^2 = 1, rs = sr^{-1} \rangle$ .

Show that every element of $D_{2n}$ which is not a power of $r$ has order 2. Deduce that $D_{2n}$ is generated by the two elements $s$ and $sr$ , both of which have order 2.

Let $x \in D_{2n}$ be a non power of $r$ . By a previous exercise, we know that $rx = xr^{-1}$ . Moreover, $x = sr^i$ for some $0 \leq i < n$ . Thus $x^2 = sr^isr^i = ssr^{-i}r^i = 1$ , hence $|x| = 2$ .