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Definition (The Dihedral Groups $D_{n}$)
For each integer $n\geq 2$, we define the dihedral group on $n$ vertices, written $D_{n}$, to be automorphism group of the regular $n$-gon. This group consists of $n$ rotations and $n$ reflections, so has order $2n$.
In some texts, the author uses $D_{2n}$ instead of $D_{n}$, at which point they'll call it the dihedral group of order $2n$ instead of the dihedral group on $n$ vertices.
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