# Paige loop

From Groupprops

## Definition

The **Paige loop** of a field is defined as follows:

- Let be the set of invertible elements in the split octonion algebra over , made into a Moufang loop by the multiplication in the algebra.
- Let be the center of . can also be defined as the subgroup of comprising . Note that has size one if has characteristic two and size two otherwise.
- The Paige loop of is defined as the quotient loop.

The Paige loop of any field is a simple Moufang loop, and it is always non-commutative and non-associative, so Paige loops are examples of simple Moufang loops that are not simple groups. For finite orders, these are the *only* examples of simple Moufang loops that are not simple groups.

The order (i.e., size) of the Paige loop over a finite field of size is given by:

## Particular cases

Field | Size of field | Characteristic of field | Paige loop | Size of Paige loop = where |
---|---|---|---|---|

field:F2 | 2 | 2 | Paige loop of field:F2 | 120 |

field:F3 | 3 | 3 | Paige loop of field:F3 | 1080 |

field:F4 | 4 | 2 | Paige loop of field:F4 | 16320 |