# Difference between revisions of "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.

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

## 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