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

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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}> | ||

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+ | ==== Reference ==== | ||

+ | 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;