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 invers elements / relation if the invers 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 relation 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;