Difference between revisions of "ApCoCoA-1:Dicyclic groups"
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| − | === <div id="Dicyclic_groups">[[:ApCoCoA:Symbolic data#Dicyclic_groups|Dicyclic | + | === <div id="Dicyclic_groups">[[:ApCoCoA:Symbolic data#Dicyclic_groups|Dicyclic Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
The dicyclic groups are non-abelian groups with order 4n. For n = 2 the dicyclic group is isomporphic to the quarternion group Q. | The dicyclic groups are non-abelian groups with order 4n. For n = 2 the dicyclic group is isomporphic to the quarternion group Q. | ||
| Line 5: | Line 5: | ||
Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}> | Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}> | ||
| − | + | ==== Reference ==== | |
| + | Coxeter, H. S. M., "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University, 1974. | ||
==== Computation ==== | ==== Computation ==== | ||
| Line 13: | Line 14: | ||
// Number of Dicyclic group (note that the order is 4N) | // Number of Dicyclic group (note that the order is 4N) | ||
MEMORY.N:=5; | MEMORY.N:=5; | ||
| − | |||
Use ZZ/(2)[a,b]; | Use ZZ/(2)[a,b]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsDicyclic() | Define CreateRelationsDicyclic() | ||
Relations:=[]; | Relations:=[]; | ||
| − | + | ||
| − | // | + | // Add the relation a^{2n} = 1 |
| + | Append(Relations, [[a^(2*MEMORY.N)], [1]]); | ||
| + | |||
| + | // Add the relation a^{n} = b^2 | ||
Append(Relations, [[a^(MEMORY.N)], [-b,b]]); | Append(Relations, [[a^(MEMORY.N)], [-b,b]]); | ||
| − | |||
| − | |||
| − | |||
| − | // | + | // Add the relation b^{-1}ab = a^{-1} |
Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]); | Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]); | ||
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
| + | |||
| + | Relations:=CreateRelationsDicyclic(); | ||
| + | Relations; | ||
| + | |||
| + | // Compute a Groebner basis | ||
| + | Gb:=NC.GB(Relations); | ||
| + | Gb; | ||
| + | |||
| + | // Compute the values of the Hilbert-Dehn function | ||
| + | NC.HF(Gb); | ||
| + | // The order of the dicyclic group | ||
| + | Sum(It); | ||
| + | |||
| + | ====Example in Symbolic Data Format==== | ||
| + | <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | ||
| + | <vars>a,b</vars> | ||
| + | <basis> | ||
| + | <ncpoly>a^(2*5)-1</ncpoly> | ||
| + | <ncpoly>a^5-b*b</ncpoly> | ||
| + | <ncpoly>b*b*b*a*b-a^(2*5-1)</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>Dicyclic_group_5</Comment> | ||
| + | </FREEALGEBRA> | ||
Latest revision as of 20:29, 22 April 2014
Description
The dicyclic groups are non-abelian groups with order 4n. For n = 2 the dicyclic group is isomporphic to the quarternion group Q. Note that every element of this groups can be written uniquely as a^k x^j for 0 < k < 2n and j = 0 or 1.
Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}>
Reference
Coxeter, H. S. M., "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University, 1974.
Computation
/*Use the ApCoCoA package ncpoly.*/
// Number of Dicyclic group (note that the order is 4N)
MEMORY.N:=5;
Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");
Define CreateRelationsDicyclic()
Relations:=[];
// Add the relation a^{2n} = 1
Append(Relations, [[a^(2*MEMORY.N)], [1]]);
// Add the relation a^{n} = b^2
Append(Relations, [[a^(MEMORY.N)], [-b,b]]);
// Add the relation b^{-1}ab = a^{-1}
Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]);
Return Relations;
EndDefine;
Relations:=CreateRelationsDicyclic();
Relations;
// Compute a Groebner basis
Gb:=NC.GB(Relations);
Gb;
// Compute the values of the Hilbert-Dehn function
NC.HF(Gb);
// The order of the dicyclic group
Sum(It);
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a,b</vars> <basis> <ncpoly>a^(2*5)-1</ncpoly> <ncpoly>a^5-b*b</ncpoly> <ncpoly>b*b*b*a*b-a^(2*5-1)</ncpoly> </basis> <Comment>Dicyclic_group_5</Comment> </FREEALGEBRA>
