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Paige loop

411 bytes added, 06:59, 20 August 2011
Definition
* The Paige loop of <math>K</math> is defined as the [[quotient loop]]<math>M(K)/Z</math>.
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 <math>q</math> is given by: <math>\!\frac{1}{d}q^3(q^4 - 1), \qquad \operatorname{where} \qquad d = \operatorname{gcd}(2,q-1)</math>
The order (i.e., size) of the Paige loop over a finite field of size <math>q</math> is given by ?
==Particular cases==
{| class="sortable" border="1"
! 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:F3]] || 3 || 3 || ? [[Paige loop of field:F3]] || ?1080
|-
| [[field:F4]] || 4|| 2 || ? [[Paige loop of field:F4]] || ?16320
|}
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