Difference between revisions of "ApCoCoA-1:Dihedral groups"

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==== Description ====
 
==== Description ====
 
The dihedral group of degree n is the group of symmetries of a regular polynom. This non-abelian group consists of 2n elements, n rotations and n reflections. Let r be a single rotation and s be an arbitrary reflection. Then the group has the following representation
 
The dihedral group of degree n is the group of symmetries of a regular polynom. This non-abelian group consists of 2n elements, n rotations and n reflections. Let r be a single rotation and s be an arbitrary reflection. Then the group has the following representation
  Dih(n) = <r,s | r^{n} = s^{2} = s^{-1}rs = r^{-1}>
+
  Dih(n) = <r,s | r^{n} = s^{2} = s^{-1}rs = r^{-1} = 1>
  
(Reference: Reflection Groups and Invariant Theory, Richard Kane, Springer 2001)
+
==== Reference ====
 +
Reflection Groups and Invariant Theory, Richard Kane, Springer, 2001.
  
 
==== Computation ====
 
==== Computation ====

Revision as of 08:52, 23 August 2013

Description

The dihedral group of degree n is the group of symmetries of a regular polynom. This non-abelian group consists of 2n elements, n rotations and n reflections. Let r be a single rotation and s be an arbitrary reflection. Then the group has the following representation

Dih(n) = <r,s | r^{n} = s^{2} = s^{-1}rs = r^{-1} = 1>

Reference

Reflection Groups and Invariant Theory, Richard Kane, Springer, 2001.

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Number of Dihedral group
 MEMORY.N:=5;
 
 
 Use ZZ/(2)[r,s];
 NC.SetOrdering("LLEX");
 Define CreateRelationsDehidral()
   Relations:=[];
   
   // add the relation r^{n} = 1    
   Append(Relations,[[r^MEMORY.N],[1]]);
   
   // add the relation s^2 = 1
   Append(Relations,[[s^2],[1]]);
   
   // add the relation s^{-1}rs = r^{-1}
   Append(Relations,[[s,r,s],[r^(MEMORY.N-1)]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsDehidral();
 Relations;
 GB:=NC.GB(Relations);
 GB;