Order of Dihedral Group
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Let $P$ be a regular $n$-gon.
By inspection, it is seen that:
- $(1): \quad$ there are $n$ symmetries of the vertices of $P$ by rotation
- $(2): \quad$ there are a further $n$ symmetries of the vertices of $P$ by rotation after reflected in any of the axes of symmetry of $P$.
Hence the result.