Difference between revisions of "ApCoCoA-1:Dihedral groups"
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(New page: === <div id="Dihedral groups">Dihedral groups</div> === ==== Description ==== The dihedral group of degree n is the group of symmetries of a regu...) |
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| − | === <div id="Dihedral groups">[[:ApCoCoA:Symbolic data#Dihedral_groups|Dihedral | + | === <div id="Dihedral groups">[[:ApCoCoA:Symbolic data#Dihedral_groups|Dihedral Groups]]</div> === |
==== 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 ==== | ||
| Line 12: | Line 13: | ||
// Number of Dihedral group | // Number of Dihedral group | ||
MEMORY.N:=5; | MEMORY.N:=5; | ||
| − | |||
Use ZZ/(2)[r,s]; | Use ZZ/(2)[r,s]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| − | Define | + | |
| + | Define CreateRelationsDihedral() | ||
Relations:=[]; | Relations:=[]; | ||
| − | // add the relation r^{n} = 1 | + | // add the relation r^{n} = 1 |
| − | |||
Append(Relations,[[r^MEMORY.N],[1]]); | Append(Relations,[[r^MEMORY.N],[1]]); | ||
| Line 32: | Line 32: | ||
EndDefine; | EndDefine; | ||
| − | Relations:= | + | Relations:=CreateRelationsDihedral(); |
Relations; | 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> | ||
Latest revision as of 20:29, 22 April 2014
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>
