Problem

\(D_{2n}=\langle r,s | r^n = s^2 = 1, rs = sr^{-1}\rangle\).

Use the generators and relations above to show that if \(x\) is any element of \(D_{2n}\) which is not a power of \(r\), then \(rx=xr^{-1}\).

Solution

Since \(x\) is not a power of \(r\), \(x = sr^i\) for some \(0 \le i < n\).

Using one of the given relations \(rs = sr^{-1}\),

$$ rx = r(sr^i) = (rs)r^i = (sr^{-1})r^i = sr^{i-1} = (sr^i)r^{-1} = xr^{-1} $$

$$\tag*{$\blacksquare$}$$