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

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(New page: === <div id="Fibonacci_groups">Fibonacci groups</div> === ==== Description ==== Fibonacci groups are related to the inductive definition of the ...)
 
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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).
 
   F(2,m) = <x_{1},...,x_{m} | x_{i}x_{i+1} = x_{i+2}>
 
   F(2,m) = <x_{1},...,x_{m} | x_{i}x_{i+1} = x_{i+2}>
(Reference: R. Thomas, “The Fibonacci groups F(2,2m),”Bull. London Math. Soc.,21, No. 5, 463-465 (1989)
 
  
 
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   </tr>
 
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==== Reference ====
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R. Thomas, “The Fibonacci groups F(2,2m)”, Bull. London Math. Soc.,21, No. 5, 463-465 (1989).
  
 
==== Computation ====
 
==== Computation ====

Revision as of 09:20, 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 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;