Difference between revisions of "Paige loop"

From Groupprops
Jump to: navigation, search
(Particular cases)
Line 13: Line 13:
  
 
{| class="sortable" border="1"
 
{| class="sortable" border="1"
! Field !! Size of field !! Characteristic of field !! Paige loop !! Size of Paige loop
+
! Field !! Size of field <math>q</math> !! Characteristic of field <math>p</math> !! Paige loop !! Size of Paige loop = <math>\frac{1}{d}q^3(q^4 - 1)</math> where <math>d = \operatorname{gcd}(2,q - 1)</math>
 
|-
 
|-
 
| [[field:F2]] || 2 || 2 || [[Paige loop of field:F2]] || 120
 
| [[field:F2]] || 2 || 2 || [[Paige loop of field:F2]] || 120
 
|-
 
|-
| [[field:F3]] || 3 || 3 || ? || ?
+
| [[field:F3]] || 3 || 3 || [[Paige loop of field:F3]] || 1080
 
|-
 
|-
| [[field:F4]] || 4|| 2 || ? || ?
+
| [[field:F4]] || 4|| 2 || [[Paige loop of field:F4]] || 16320
 
|}
 
|}

Revision as of 03:45, 20 August 2011

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 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