Difference between revisions of "Dihedral group"

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{{natural number-parametrized group family}}
 
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{{particularcases|[[:Category:Dihedral groups]]}}
 
==Definition==
 
==Definition==
  

Revision as of 23:54, 2 January 2008

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dicyclic group (also called binary dihedral group)
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family

This article is about a general term. A list of important particular cases (instances) is available at Category:Dihedral groups

Definition

The dihedral group with parameter n, denoted sometimes as D_n and sometimes as D_{2n} is defined in the following equivalent ways:

<x,a|a^n = x^2 = 1, xax^{-1} = a^{-1}>

  • It is the group of symmetries of a regular n-gon in the plane, viz the plane isometries that preserves the set of points of the regular n-gon.