Difference between revisions of "Dihedral group"
From Encyclopedia of Mathematics
m (Richard Pinch moved page Dihedron group to Dihedral group: more common name) |
(alternative description is the group of rotations and reflections of a regular polygon) |
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| − | '' | + | ''dihedron group'' |
| − | A group isomorphic to the rotation group of a dihedron, i.e. of a regular doubled [[ | + | 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. |
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| + | 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. | ||
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| + | See also: [[Quasi-dihedral group]] | ||
====References==== | ====References==== | ||
Revision as of 09:00, 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.
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) |
How to Cite This Entry:
Dihedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dihedral_group&oldid=33921
Dihedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dihedral_group&oldid=33921
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article