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Difference between revisions of "Dihedral group"

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(alternative description is the group of rotations and reflections of a regular polygon)
(Comment on notation)
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''dihedron 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.
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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.  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. Two different elements of order 2 in any finite group generate a dihedral group.
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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)
How to Cite This Entry:
Dihedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dihedral_group&oldid=33924
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article