Difference between revisions of "Fibonacci group"

Latest revision as of 20:45, 3 September 2009

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 $m$ .

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

@@ Line 3: / Line 3: @@
 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"
+! <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>
+|-
+| <math>4</math> || [[Cyclic group:Z5]] || <math>5</math>
+|-
+| <math>5</math> || [[Cyclic group:Z11]] || <math>11</math>
+|-
+| <math>7</math> || [[Cyclic group:Z29]] || <math>29</math>
+|}
 ==External links==
 * {{sor|F/f130070}}