Difference between revisions of "Paige loop"
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* The Paige loop of <math>K</math> is defined as the [[quotient loop]]<math>M(K)/Z</math>. | * 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. | + | 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. |
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| + | The order (i.e., size) of the Paige loop over a finite field of size <math>q</math> is given by: | ||
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| + | <math>\!\frac{1}{d}q^3(q^4 - 1), \qquad \operatorname{where} \qquad d = \operatorname{gcd}(2,q-1)</math> | ||
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==Particular cases== | ==Particular cases== | ||
{| 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 || | + | | [[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 |
|} | |} | ||
Latest revision as of 06:59, 20 August 2011
Definition
The Paige loop of a field 
is defined as follows:
- Let
be the set of invertible elements in the split octonion algebra over , made into a Moufang loop by the multiplication in the algebra. - Let
be the center of . can also be defined as the subgroup of comprising 
. Note that has size one if has characteristic two and size two otherwise. - The Paige loop of is defined as the quotient loop
.
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 
is given by:
Particular cases
| Field | Size of field |
Characteristic of field  |
Paige loop | Size of Paige loop =  where 
|
|---|---|---|---|---|
| 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 |
