Paige loop

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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 loopM(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:

\!\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