Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 1D (Groups)

The centre isEvery subgroup $H\le Z(G)$ is normal, since $gh{g}^{-1}=h$ for all $g\in G$ and $h\in H$.

Write $D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$. A central rotation must satisfy $r^k=r^{-k}$, and no reflection is central for $n\ge 3$. Hence