# 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. |

+ | |||

+ | 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> | ||

− | |||

==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 |