# Difference between revisions of "Loop of order five and exponent two"

From Groupprops

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<math>\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 4 & 5 & 3 \\ 3 & 5 & 1 & 2 & 4 \\ 4 & 3 & 5 & 1 & 2 \\ 5 & 4 & 2 & 3 & 1 \\\end{pmatrix}</math> | <math>\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 4 & 5 & 3 \\ 3 & 5 & 1 & 2 & 4 \\ 4 & 3 & 5 & 1 & 2 \\ 5 & 4 & 2 & 3 & 1 \\\end{pmatrix}</math> | ||

− | == | + | ==Loop properties== |

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

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| [[dissatisfies property::Right alternative loop]] || No || For instance, <math>2 * (3 * 3) \ne (2 * 3) * 3</math> | | [[dissatisfies property::Right alternative loop]] || No || For instance, <math>2 * (3 * 3) \ne (2 * 3) * 3</math> | ||

|- | |- | ||

− | | [dissatisfies property::Alternative loop]] || No || Neither left nor right alternative | + | | [[dissatisfies property::Alternative loop]] || No || Neither left nor right alternative |

|- | |- | ||

| [[dissatisfies property::Left Bol loop]] || No || Not left alternative, so not left Bol | | [[dissatisfies property::Left Bol loop]] || No || Not left alternative, so not left Bol | ||

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|- | |- | ||

| [[dissatisfies property::C-loop]] || No || Not alternative, so not C. | | [[dissatisfies property::C-loop]] || No || Not alternative, so not C. | ||

+ | |- | ||

+ | | [[dissatisfies property::Monogenic loop]] || No || The subloop generated by every element is a subgroup of order two. | ||

|} | |} |

## Latest revision as of 15:44, 26 June 2012

This article is about a particular loop, viz a loop unique upto isomorphism

View a complete list of particular loops

## Definition

This loop is defined by the multiplication table:

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

1 | 1 | 2 | 3 | 4 | 5 |

2 | 2 | 1 | 4 | 5 | 3 |

3 | 3 | 5 | 1 | 2 | 4 |

4 | 4 | 3 | 5 | 1 | 2 |

5 | 5 | 4 | 2 | 3 | 1 |

In other words, this is the algebra loop corresponding to the Latin square:

## Loop properties

Property | Satisfied? | Explanation |
---|---|---|

Loop of exponent two | Yes | Every non-identity element squares to the identity |

Power-associative loop | Yes | Follows since it is a loop of exponent two |

Flexible loop | Yes | In fact, for all |

Commutative loop | No | For instance, |

Left alternative loop | No | For instance, |

Right alternative loop | No | For instance, |

Alternative loop | No | Neither left nor right alternative |

Left Bol loop | No | Not left alternative, so not left Bol |

Right Bol loop | No | Not right alternative, so not right Bol |

Moufang loop | No | Not alternative, so not Moufang |

Left nuclear square loop | Yes | Every square is the identity element, hence in the left nucleus |

Middle nuclear square loop | Yes | Every square is the identity element, hence in the middle nucleus |

Right nuclear square loop | Yes | Every square is the identity element, hence in the right nucleus |

LC-loop | No | Not left alternative, so not LC. |

RC-loop | No | Not right alternative, so not RC. |

C-loop | No | Not alternative, so not C. |

Monogenic loop | No | The subloop generated by every element is a subgroup of order two. |