# 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. | ||

==== 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;