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