ApCoCoA-1:Fibonacci groups

Fibonacci Groups

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>