# Loop of order five and exponent two

From Groupprops

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. |