Dihedral group

dihedron group

A finite group isomorphic to the rotation group of a dihedron, i.e. of a regular doubled pyramid. If the base of the pyramid is an $n$-gon, the corresponding dihedron group is of order $2n$ and is generated by two rotations $\phi$ and $\psi$ of orders $n$ and $2$ respectively, with the defining relation $\phi\psi\phi\psi=1$. An alternative description is the group of rotations and reflections of a regular $n$-gon, with $\phi$ as a rotation of order $n$ and $\psi$ as a reflection.

A dihedral group is sometimes understood to denote the dihedral group of order 8 only. Two different elements of order 2 in any finite group generate a dihedral group.

See also: Quasi-dihedral group

References

[1]	G.G. Hall, "Applied group theory" , Longman (1967)