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Dihedral group

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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. In a finite group, two different elements of order 2 generate a dihedral group.

The notation for the group varies: some author write $D_n$ and others $D_{2n}$ for the group of order $2n$. A dihedral group is sometimes understood to denote the dihedral group of order 8 only.

The dihedral group is the semi-direct product of cyclic groups $C_2$ by $C_n$, with $C_2$ acting on $C_n$ by the non-trivial element of $C_2$ mapping each element of $C_n$ to its inverse.

See also: Quasi-dihedral group

References

[1] G.G. Hall, "Applied group theory" , Longman (1967)
How to Cite This Entry:
Dihedron group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dihedron_group&oldid=33922