Difference between revisions of "Fibonacci group"

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The '''Fibonacci group''' <math>F(2,m)</math> is defined as the group with the following [[presentation]]:
 
The '''Fibonacci group''' <math>F(2,m)</math> is defined as the group with the following [[presentation]]:
  
<math>\{ x_1, x_2, \ldots, x_m | x_ix_{i+1} = x_{i+2}\}</math>
+
<math>\{ x_1, x_2, \dots, x_m | x_ix_{i+1} = x_{i+2}\}</math>
  
where the indices are reduced modulo <math>m</math>.
+
where the indexes are reduced modulo <math>m</math>.
 +
 
 +
==Particular cases==
 +
 
 +
The only cases where <math>F(2,m)</math> is finite are <math>m = 1,2,3,4,5,7</math>:
 +
 
 +
{| class="wikitable" border="1"
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! <math>m</math> !! Common name for <math>F(2,m)</math> !! Order of group
 +
|-
 +
| <math>1</math> || [[Trivial group]] || <math>1</math>
 +
|-
 +
| <math>2</math> || [[Trivial group]] || <math>1</math>
 +
|-
 +
| <math>3</math> || [[Quaternion group]] || <math>8 = 2^3</math>
 +
|-
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| <math>4</math> || [[Cyclic group:Z5]] || <math>5</math>
 +
|-
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| <math>5</math> || [[Cyclic group:Z11]] || <math>11</math>
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|-
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| <math>7</math> || [[Cyclic group:Z29]] || <math>29</math>
 +
|}
  
 
==External links==
 
==External links==
  
 
* {{sor|F/f130070}}
 
* {{sor|F/f130070}}

Latest revision as of 20:45, 3 September 2009

Definition

The Fibonacci group F(2,m) is defined as the group with the following presentation:

\{ x_1, x_2, \dots, x_m | x_ix_{i+1} = x_{i+2}\}

where the indexes are reduced modulo m.

Particular cases

The only cases where F(2,m) is finite are m = 1,2,3,4,5,7:

m Common name for F(2,m) Order of group
1 Trivial group 1
2 Trivial group 1
3 Quaternion group 8 = 2^3
4 Cyclic group:Z5 5
5 Cyclic group:Z11 11
7 Cyclic group:Z29 29

External links