Sylow \(p\)-subgroups of \(D_{2n}\) (D&F 4.5.5)

Show that a Sylow \(p\)-subgroup of \(D_{2n}\) is cyclic and normal for every odd prime \(p\).

Let \(2n = p^\alpha m\), where \(p \nmid m\). Note that \(m\) is even since \(2n\) is even and \(p^\alpha\) is odd. Thus we can consider the cyclic subgroup \(\langle r^{m/2} \rangle\), which has order \(p^\alpha\) and is hence a Sylow \(p\)-subgroup. Furthermore, \(\langle r^{m/2} \rangle \trianglelefteq D_{2n}\) so it is unique.