## For odd primes p, the Sylow p-subgroups of Dih(2n) are cyclic and normal

Prove that for any odd prime $p$, a Sylow $p$-subgroup of $D_{2n}$ is cyclic and normal in $D_{2n}$.

Let $P \leq D_{2n}$ be a Sylow $p$-subgroup, where $p$ is an odd prime. Note that every element of the form $sr^a$ has order 2 and thus by Lagrange’s Theorem cannot be an element of $P$; thus $P \leq \langle r \rangle$. Hence $P$ is cyclic. Moreover, in this previous exercise we showed that every subgroup of $\langle r \rangle$ (in particular $P$) is normal in $D_{2n}$.