Difference between revisions of "ApCoCoA-1:Fibonacci groups"

From ApCoCoAWiki
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 invers elements / relation if the invers elements
+
 
 +
     // 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,[[x[Index1],y[Index1]],[1]]);
 
       //Append(Relations,[[y[Index1],x[Index1]],[1]]);
 
       //Append(Relations,[[y[Index1],x[Index1]],[1]]);
 
     EndFor;
 
     EndFor;
     // add the relation x_{i}x_{i+1} = x_{i+2}
+
 
 +
     // 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;
+
  Gb:=NC.GB(Relations,31,1,100,1000);
 +
   Gb;

Revision as of 09:21, 23 August 2013

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;