Dihedral group D_2 - Knowlpedia

the dihedral group $D_{2}$ is the semidirect product $C_1\rtimes C_2$ in which the nontrivial element of $C_2$ acts by inversion. Equivalently, its elements are pairs $(a,\epsilon)\in(\mathbb Z/1\mathbb Z)\times(\mathbb Z/2\mathbb Z)$ , with

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

The coordinates are reduced modulo $1$ and $2$ , respectively; the subscript records the group order.

Order and simplicity

The group has $2$ elements. When $n=1$ , the first factor is trivial, so this is $C_2$ , which is simple.

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).