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 ?

Particular cases

Field Size of field Characteristic of field Paige loop Size of Paige loop
field:F2 2 2  ? 120
field:F3 3 3  ?  ?
field:F4 4 2  ?  ?