Difference between revisions of "Dihedral group"

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* It is the group of symmetries of a regular <math>n</math>-gon in the plane, viz the plane isometries that preserves the set of points of the regular <math>n</math>-gon.
 
* It is the group of symmetries of a regular <math>n</math>-gon in the plane, viz the plane isometries that preserves the set of points of the regular <math>n</math>-gon.
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==References==
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===Textbook references==
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* {{booklink-defined-cum-explored|DummitFoote}}, Page 23-27, Section 1.2 ''Dihedral Groups'' (the entire section discusses dihedral groups from a number of perspectives)
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* {{booklink-defined|AlperinBell}}, Page 24 (definition introduced in paragraph)
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* {{booklink-defined|Lang}}, Page 78, Exercise 34 (a) (definition introduced in exercise)
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* {{booklink-defined|Herstein}}, Page 54, Problem 17
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* {{booklink-defined|RobinsonGT}}, Page 6 (definition introduced informally, in paragraph, using the geometric perspective)
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* {{booklink-defined|RobinsonAA}}, Page 42, under ''The symmetry group of the regular n-gon''
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* {{booklink-defined|Hungerford}}, Page 50 (definition introduced as a subgroup of the [[symmetric group]])

Revision as of 13:06, 25 May 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.

References

=Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info, Page 23-27, Section 1.2 Dihedral Groups (the entire section discusses dihedral groups from a number of perspectives)
  • Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, More info, Page 24 (definition introduced in paragraph)
  • Algebra by Serge Lang, ISBN 038795385X, More info, Page 78, Exercise 34 (a) (definition introduced in exercise)
  • Topics in Algebra by I. N. Herstein, More info, Page 54, Problem 17
  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 6 (definition introduced informally, in paragraph, using the geometric perspective)
  • An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, More info, Page 42, under The symmetry group of the regular n-gon
  • Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, More info, Page 50 (definition introduced as a subgroup of the symmetric group)