Difference between revisions of "ApCoCoA-1:Fibonacci groups"
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| − | === <div id="Fibonacci_groups">[[:ApCoCoA:Symbolic data#Fibonacci_groups|Fibonacci | + | === <div id="Fibonacci_groups">[[:ApCoCoA:Symbolic data#Fibonacci_groups|Fibonacci Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
Fibonacci groups are related to the inductive definition of the Fibonacci numbers f_{i} + f_{i+1} = f_{i+2} where f_{1} = f_{2} = 1. For a natural number m <= 7 this groups are finite (see table below). | Fibonacci groups are related to the inductive definition of the Fibonacci numbers f_{i} + f_{i+1} = f_{i+2} where f_{1} = f_{2} = 1. For a natural number m <= 7 this groups are finite (see table below). | ||
| Line 49: | Line 49: | ||
// Number of fibonacci group | // Number of fibonacci group | ||
| − | |||
MEMORY.N:=7; | MEMORY.N:=7; | ||
Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsFibonacci() | Define CreateRelationsFibonacci() | ||
Relations:=[]; | Relations:=[]; | ||
| − | // add the | + | |
| + | // add the relations of the inverse elements | ||
For Index1 := 1 To MEMORY.N Do | For Index1 := 1 To MEMORY.N Do | ||
| − | + | Append(Relations,[[x[Index1],y[Index1]],[1]]); | |
| − | + | Append(Relations,[[y[Index1],x[Index1]],[1]]); | |
EndFor; | EndFor; | ||
| − | // add the | + | |
| + | // add the relations x_{i}x_{i+1} = x_{i+2} | ||
For Index2 := 1 To MEMORY.N -2 Do | For Index2 := 1 To MEMORY.N -2 Do | ||
Append(Relations,[[x[Index2],x[Index2+1]],[x[Index2+2]]]); | Append(Relations,[[x[Index2],x[Index2+1]],[x[Index2+2]]]); | ||
| Line 67: | Line 69: | ||
Append(Relations,[[x[MEMORY.N-1],x[MEMORY.N]],[x[1]]]); | Append(Relations,[[x[MEMORY.N-1],x[MEMORY.N]],[x[1]]]); | ||
Append(Relations,[[x[MEMORY.N],x[1]],[x[2]]]); | Append(Relations,[[x[MEMORY.N],x[1]],[x[2]]]); | ||
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
| Line 72: | Line 75: | ||
Relations:=CreateRelationsFibonacci(); | Relations:=CreateRelationsFibonacci(); | ||
Relations; | Relations; | ||
| − | + | ||
| − | + | Gb:=NC.GB(Relations,31,1,100,1000); | |
| + | Gb; | ||
| + | |||
| + | ====Example in Symbolic Data Format==== | ||
| + | <FREEALGEBRA createdAt="2014-03-11" createdBy="strohmeier"> | ||
| + | <vars>x1,x2,x3,x4,x5,x6,x7</vars> | ||
| + | <uptoDeg>6</uptoDeg> | ||
| + | <basis> | ||
| + | <ncpoly>x1*x2-x3</ncpoly> | ||
| + | <ncpoly>x2*x3-x4</ncpoly> | ||
| + | <ncpoly>x3*x4-x5</ncpoly> | ||
| + | <ncpoly>x4*x5-x6</ncpoly> | ||
| + | <ncpoly>x5*x6-x7</ncpoly> | ||
| + | <ncpoly>x6*x7-x1</ncpoly> | ||
| + | <ncpoly>x7*x1-x2</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>The partial LLex Gb has 423 elements</Comment> | ||
| + | <Comment>Fibonacci_group_7</Comment> | ||
| + | </FREEALGEBRA> | ||
Latest revision as of 20:48, 22 April 2014
Description
Fibonacci groups are related to the inductive definition of the Fibonacci numbers f_{i} + f_{i+1} = f_{i+2} where f_{1} = f_{2} = 1. For a natural number m <= 7 this groups are finite (see table below).
F(2,m) = <x_{1},...,x_{m} | x_{i}x_{i+1} = x_{i+2}>
| m | isomorphic group | order |
| 1 | trivial group | 1 |
| 2 | trivial group | 1 |
| 3 | Quaternion group | 8 |
| 4 | cyclic group Z5 | 5 |
| 5 | cyclic group Z11 | 11 |
| 7 | cyclic group Z29 | 29 |
Reference
R. Thomas, “The Fibonacci groups F(2,2m)”, Bull. London Math. Soc.,21, No. 5, 463-465 (1989).
Computation
/*Use the ApCoCoA package ncpoly.*/
// Number of fibonacci group
MEMORY.N:=7;
Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
NC.SetOrdering("LLEX");
Define CreateRelationsFibonacci()
Relations:=[];
// add the relations of the inverse elements
For Index1 := 1 To MEMORY.N Do
Append(Relations,[[x[Index1],y[Index1]],[1]]);
Append(Relations,[[y[Index1],x[Index1]],[1]]);
EndFor;
// add the relations x_{i}x_{i+1} = x_{i+2}
For Index2 := 1 To MEMORY.N -2 Do
Append(Relations,[[x[Index2],x[Index2+1]],[x[Index2+2]]]);
EndFor;
Append(Relations,[[x[MEMORY.N-1],x[MEMORY.N]],[x[1]]]);
Append(Relations,[[x[MEMORY.N],x[1]],[x[2]]]);
Return Relations;
EndDefine;
Relations:=CreateRelationsFibonacci();
Relations;
Gb:=NC.GB(Relations,31,1,100,1000);
Gb;
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-11" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,x7</vars> <uptoDeg>6</uptoDeg> <basis> <ncpoly>x1*x2-x3</ncpoly> <ncpoly>x2*x3-x4</ncpoly> <ncpoly>x3*x4-x5</ncpoly> <ncpoly>x4*x5-x6</ncpoly> <ncpoly>x5*x6-x7</ncpoly> <ncpoly>x6*x7-x1</ncpoly> <ncpoly>x7*x1-x2</ncpoly> </basis> <Comment>The partial LLex Gb has 423 elements</Comment> <Comment>Fibonacci_group_7</Comment> </FREEALGEBRA>
