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Let $D_n$ be the dihedral group of order $2n>4$; so it contains a cyclic subgroup $C$ of order $n$ on which $\sigma \in D_n$ outside of $C$ acts as $\sigma c \sigma^{-1}=c^{-1}$ for all $c \in C$. When is the cyclic subgroup $C$ with the above property unique? Determine all $n$ for which $D_n$ has a unique cyclic subgroup $C$ and justify your answer.

Isn't it true that all $D_n$ have a unique cyclic subgroup of order $n$, namely the one that includes all the rotations? And isn't it true that the reflections act the way described above on the rotations? But then the problem seems trivial, so perhaps I'm misunderstanding something?

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You are right, the group $D_n$ contains exactly one cyclic subgroup of order $n$ when $n > 2$. To prove this, you can use the fact that every nonidentity element outside $C$ has order $2$.

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