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}.

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