ApCoCoA-1:Dihedral groups

Dihedral Groups

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 CreateRelationsDihedral()
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:=CreateRelationsDihedral();
Relations;

Gb:=NC.GB(Relations);
Gb;
```

Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>r,s</vars>
<basis>
<ncpoly>r^5-1</ncpoly>
<ncpoly>s*s-1</ncpoly>
<ncpoly>s*r*s-r^(5-1)</ncpoly>
</basis>
<Comment>Dihedral_group_5</Comment>
</FREEALGEBRA>
```