Exercise 1703Algebra › GroupsL1Added 3 Nov 2026

The dihedral group $D_n$

#dihedral group#symmetries#presentation

Statement

Describe the dihedral group $D_n$ (symmetries of a regular $n$ -gon), give its order, and a presentation.

Hint

Symmetries are generated by a rotation $r$ of order $n$ and a reflection $s$ of order $2$ , with $srs=r^{-1}$ .

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Order. $D_n$ has $2n$ elements: $n$ rotations $r^0,\ldots,r^{n-1}$ and $n$ reflections $sr^k$ ( $k=0,\ldots,n-1$ ).

Presentation. $\langle r,s:r^n=s^2=e,\,srs=r^{-1}\rangle$ .

Example $n=3$ : $D_3\cong S_3$ .

Conclusion. $|D_n|=2n$ , and $D_n$ has a cyclic subgroup of rotations (normal, index $2$ ) and $n$ reflections.