Difference between revisions of "Dihedral group"
(alternative description is the group of rotations and reflections of a regular polygon) |
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| − | 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 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. |
| − | A dihedral group is sometimes understood to denote the dihedral group of order 8 only | + | 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. |
See also: [[Quasi-dihedral group]] | See also: [[Quasi-dihedral group]] | ||
Revision as of 09:04, 19 October 2014
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.
See also: Quasi-dihedral group
References
| [1] | G.G. Hall, "Applied group theory" , Longman (1967) |
Dihedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dihedral_group&oldid=33924