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Difference between revisions of "Dihedral group"

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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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''dihedral group''
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''dihedron group''
  
A group isomorphic to the rotation group of a dihedron, i.e. of a regular doubled [[Pyramid|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 $\phi\psi\phi\psi=1$. 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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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=33923
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article