Dihedral group D_2n - Knowlpedia

For an integer $n\geq1$ , the dihedral group $D_{2n}$ is the semidirect product $C_n\rtimes C_2$ in which the nontrivial element of $C_2$ acts by inversion. Equivalently, its elements are pairs $(a,\epsilon)\in(\mathbb Z/n\mathbb Z)\times(\mathbb Z/2\mathbb Z)$ , with

(a,\epsilon)(b,\delta)=(a+(-1)^\epsilon b,\epsilon+\delta).

The coordinates are reduced modulo $n$ and $2$ , respectively; the subscript records the group order.

Order and simplicity

The group has $2n$ elements. The rotation subgroup $C_n\times\{0\}$ is normal of index two. It is proper nontrivial when $n\geq2$ , while $n=1$ gives $C_2$ . Thus simplicity occurs exactly for $n=1$ .

Notation and small cases

This catalogue uses $D_{2n}$ for a group of order $2n$ ; another common convention calls the same group $D_n$ . The abstract definition admits $n=1,2$ : it gives $C_2$ and the Klein four-group, respectively. For $n\geq3$ , it is the symmetry group of a regular $n$ -gon.

References

J. S. Milne, Group Theory, v4.01, §1.17, pp. 13–14: dihedral groups (there denoted D_n with order 2n).