Paige loop

Definition

The Paige loop of a field $K$ is defined as follows:

• Let $M(K)$ be the set of invertible elements in the split octonion algebra over $K$, made into a Moufang loop by the multiplication in the algebra.
• Let $Z$ be the center of $M(K)$. $Z$ can also be defined as the subgroup of $M(K)$ comprising $\pm 1$. Note that $Z$ has size one if $K$ has characteristic two and size two otherwise.
• The Paige loop of $K$ is defined as the quotient loop$M(K)/Z$.

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 $q$ is given by:

$\!\frac{1}{d}q^3(q^4 - 1), \qquad \operatorname{where} \qquad d = \operatorname{gcd}(2,q-1)$

Particular cases

Field	Size of field $q$	Characteristic of field $p$	Paige loop	Size of Paige loop = $\frac{1}{d}q^3(q^4 - 1)$ where $d = \operatorname{gcd}(2,q - 1)$
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