# Fibonacci group

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## 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 $29$